{
  "format": "stapler-pack",
  "version": 1,
  "kind": "library",
  "meta": {
    "title": "Circle Geometry",
    "author": "Staples Education",
    "description": "A complete circle-geometry unit in six short, assignable quizzes: arcs and chords, circumference and area, equations of circles, inscribed angles, secant and tangent angles, and segment lengths. 35 procedural questions plus 15 that ask students to find the error, name the theorem, or identify what is missing.",
    "created": "2026-08-06T04:14:55.885Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "library": {
      "name": "Circle Geometry",
      "categories": [],
      "quizzes": [
        {
          "id": "circles-arcs-chords",
          "title": "Arcs, Chords & Central Angles",
          "blurb": "Central angles and the arcs they cut off, congruent chords, and the perpendicular from the centre.",
          "meta": "10 questions · fluency + reasoning",
          "categories": [
            "circles"
          ],
          "questions": [
            {
              "id": "cg-001",
              "topic": "Central Angles & Arcs",
              "difficulty": "easy",
              "stem": "In circle O, central angle ∠AOB measures 74°. What is the measure of minor arc AB?",
              "choices": [
                "37°",
                "74°",
                "148°",
                "286°"
              ],
              "correctIndex": 1,
              "explanations": [
                "37° is half of 74°, which is the inscribed-angle relationship — an angle with its vertex ON the circle intercepts an arc twice its size. This vertex is at the centre, where no halving happens.",
                "Correct. A central angle has its vertex at the centre, and its intercepted arc has exactly the same measure as the angle.",
                "148° doubles the angle. Doubling is what you do to an INSCRIBED angle to get its arc; a central angle already equals its arc.",
                "286° is the MAJOR arc AB, the long way round (360° − 74°). The question asks for the minor arc."
              ],
              "hint": "Where is the vertex of the angle — at the centre, or on the circle? That decides whether any halving or doubling applies.",
              "diagram": "<svg viewBox=\"0 0 240 150\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"74\" r=\"52\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><path d=\"M 163.1 44.9 A 52 52 0 0 0 103.9 24.5\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"4.5\" stroke-linecap=\"round\" opacity=\"0.85\"/><line x1=\"120\" y1=\"74\" x2=\"103.9\" y2=\"24.5\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"74\" x2=\"163.1\" y2=\"44.9\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"74\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"103.9\" cy=\"24.5\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"163.1\" cy=\"44.9\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"111\" y=\"78\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">O</text><text x=\"94.9\" y=\"21.5\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"173.1\" y=\"42.9\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"120\" y=\"60\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">74°</text></svg>",
              "diagramCaption": "Not drawn to scale"
            },
            {
              "id": "cg-002",
              "topic": "Central Angles & Arcs",
              "difficulty": "easy",
              "stem": "Minor arc CD in circle P measures 128°. What is the measure of central angle ∠CPD?",
              "choices": [
                "64°",
                "128°",
                "232°",
                "256°"
              ],
              "correctIndex": 1,
              "explanations": [
                "64° halves the arc. That relationship belongs to inscribed angles, whose vertex sits on the circle — not to a central angle.",
                "Correct. Central angle and intercepted arc are equal in measure, so the angle is 128°.",
                "232° is 360° − 128°, the reflex angle at P that opens onto the major arc. ∠CPD is the ordinary angle intercepting the minor arc.",
                "256° doubles the arc. Nothing in the central-angle relationship doubles; the two measures are simply equal."
              ]
            },
            {
              "id": "cg-003",
              "topic": "Central Angles & Arcs",
              "difficulty": "medium",
              "stem": "Points A, B and C lie on circle O. Arc AB = 112° and arc BC = 145°. What is the measure of arc CA (the remaining arc)?",
              "choices": [
                "93°",
                "103°",
                "115°",
                "257°"
              ],
              "correctIndex": 1,
              "explanations": [
                "93° would make the three arcs total 350°. Check the sum: the arcs of a full circle must come to exactly 360°.",
                "Correct. The three arcs make a complete circle, so arc CA = 360° − 112° − 145° = 103°.",
                "115° totals 372°, overshooting the full circle by 12°.",
                "257° is 112° + 145°, the sum of the two GIVEN arcs rather than what is left over. Subtract that sum from 360° instead."
              ],
              "hint": "Three arcs with no gaps and no overlaps make one complete trip around the circle.",
              "diagram": "<svg viewBox=\"0 0 240 154\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"76\" r=\"54\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><circle cx=\"110.6\" cy=\"22.8\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"172.8\" cy=\"87.2\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"70.3\" cy=\"97.1\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"110.6\" y=\"14.8\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"184.8\" y=\"87.2\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"58.3\" y=\"101.1\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"154\" y=\"50\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">112°</text><text x=\"122\" y=\"122\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">145°</text><text x=\"84\" y=\"56\" fill=\"#1f2024\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>"
            },
            {
              "id": "cg-004",
              "topic": "Central Angles & Arcs",
              "difficulty": "medium",
              "stem": "AC is a diameter of circle O, and B is a point on the circle. If arc AB = 63°, what is the measure of arc BC?",
              "choices": [
                "27°",
                "63°",
                "117°",
                "297°"
              ],
              "correctIndex": 2,
              "explanations": [
                "27° is 90° − 63°. Complementary angles are not involved here — a diameter cuts off a semicircle of 180°, not a right angle of 90°.",
                "63° would make arc AB and arc BC equal, which only happens if B is the midpoint of the semicircle. Nothing here says it is.",
                "Correct. A diameter splits the circle into two semicircles of 180° each, so arc BC = 180° − 63° = 117°.",
                "297° is 360° − 63°, which is the major arc from A back to B the long way. B and C are both on the same semicircle as measured here."
              ],
              "hint": "What is the arc measure of a semicircle?"
            },
            {
              "id": "cg-005",
              "topic": "Arcs & Chords",
              "difficulty": "easy",
              "stem": "In circle O, chord AB is congruent to chord CD. If arc AB = 82°, what is the measure of arc CD?",
              "choices": [
                "41°",
                "82°",
                "98°",
                "278°"
              ],
              "correctIndex": 1,
              "explanations": [
                "41° halves the arc. There is no halving in the chord–arc relationship; congruent chords cut off arcs of the same size.",
                "Correct. In the same circle, congruent chords intercept congruent arcs — so arc CD is also 82°.",
                "98° is the supplement of 82°. Supplements appear with cyclic quadrilaterals, not with congruent chords.",
                "278° is the major arc (360° − 82°). Both chords cut off minor arcs of equal measure."
              ],
              "hint": "Equal chords sit the same distance from the centre, so they subtend equal central angles."
            },
            {
              "id": "cg-006",
              "topic": "Arcs & Chords",
              "difficulty": "medium",
              "stem": "A chord of length 24 is drawn in a circle of radius 13. How far is the chord from the centre of the circle?",
              "choices": [
                "5",
                "6.5",
                "7",
                "12"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. The perpendicular from the centre bisects the chord, giving a right triangle with legs 12 and d and hypotenuse 13. Then d = √(169 − 144) = 5.",
                "6.5 is half the radius. Halving the radius has no geometric meaning here — the distance comes out of the Pythagorean relationship, not out of the radius alone.",
                "7 is 13 − 6, mixing the radius with part of the chord by subtracting. The three lengths form a right triangle, so they combine by a² + b² = c², not by subtraction.",
                "12 is half the chord, which is one LEG of the right triangle. The question asks for the other leg."
              ],
              "hint": "Drop a perpendicular from the centre to the chord. What does it do to the chord, and what triangle appears?",
              "diagram": "<svg viewBox=\"0 0 240 152\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"70\" r=\"55\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"70.2\" y1=\"93.2\" x2=\"169.8\" y2=\"93.2\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"70\" x2=\"120\" y2=\"93.2\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"70\" x2=\"169.8\" y2=\"93.2\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"70\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"70.2\" cy=\"93.2\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"169.8\" cy=\"93.2\" r=\"3.2\" fill=\"#7c3aed\"/><rect x=\"113\" y=\"85.2\" width=\"8\" height=\"8\" fill=\"none\" stroke=\"#1f2024\" stroke-width=\"1.4\"/><text x=\"60.2\" y=\"97.2\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"180.8\" y=\"97.2\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"110\" y=\"68\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">O</text><text x=\"110\" y=\"81.6\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"144.9\" y=\"75.6\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">13</text><text x=\"144.9\" y=\"108.2\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">12</text></svg>",
              "diagramCaption": "The segment from the centre meets the chord at a right angle"
            },
            {
              "id": "cg-007",
              "topic": "Arcs & Chords",
              "difficulty": "easy",
              "stem": "In circle O, diameter EF is perpendicular to chord GH, meeting it at point M. If GH = 30, what is the length of GM?",
              "choices": [
                "7.5",
                "15",
                "30",
                "60"
              ],
              "correctIndex": 1,
              "explanations": [
                "7.5 is a quarter of the chord. The perpendicular cuts the chord into two equal halves, not four quarters.",
                "Correct. A diameter perpendicular to a chord bisects it, so GM = 30 ÷ 2 = 15.",
                "30 is the whole chord GH. GM is only the piece from G to the point where the diameter crosses.",
                "60 doubles the chord. Nothing here doubles — the diameter divides the chord in two."
              ]
            },
            {
              "id": "cg-008",
              "topic": "Arcs & Chords",
              "difficulty": "hard",
              "stem": "In a circle of radius 25, two chords are each 7 units from the centre. What is the length of each chord?",
              "choices": [
                "24",
                "32",
                "48",
                "96"
              ],
              "correctIndex": 2,
              "explanations": [
                "24 is √(625 − 49) rounded incorrectly, and in any case that value is HALF the chord. √576 = 24, so the full chord is twice that.",
                "32 does not come out of the Pythagorean relationship here; check √(25² − 7²) carefully.",
                "Correct. Half the chord is √(25² − 7²) = √576 = 24, so the full chord is 48. Chords equidistant from the centre are congruent, so both measure 48.",
                "96 doubles the answer a second time. The half-chord is 24, so one doubling gives the full chord."
              ],
              "hint": "Find the half-chord with the Pythagorean theorem first, then remember which length the question is asking for."
            },
            {
              "id": "cg-041",
              "topic": "Reasoning: What's Missing",
              "difficulty": "hard",
              "stem": "A chord of a circle measures 16. A student is asked to find the chord's distance from the centre and says it cannot be done. What single additional piece of information would make it solvable?",
              "choices": [
                "The measure of the arc the chord cuts off",
                "The radius of the circle",
                "The length of a second chord",
                "Nothing else is needed; the answer is 8"
              ],
              "correctIndex": 1,
              "explanations": [
                "The arc measure would fix the shape but not the size — the same arc measure occurs in circles of every size, each giving a different distance.",
                "Correct. With the radius r, the perpendicular from the centre bisects the chord into halves of 8, and the distance is √(r² − 64). Without r there is nothing to compute.",
                "A second chord gives no new information about this one unless its distance from the centre is also known.",
                "8 is HALF THE CHORD, not the distance to the centre. Those are two different legs of the same right triangle."
              ],
              "hint": "Draw the right triangle. Which of its three sides do you already know?",
              "kind": "reasoning"
            },
            {
              "id": "cg-044",
              "topic": "Reasoning: Choose the Theorem",
              "difficulty": "medium",
              "stem": "A carpenter needs to find the exact centre of a circular tabletop. Which construction locates it, and why does it work?",
              "choices": [
                "Draw one chord and take its midpoint; the centre is always the midpoint of a chord.",
                "Draw two non-parallel chords and construct each one's perpendicular bisector; the centre is where they meet.",
                "Draw a tangent and measure inward by the radius.",
                "Draw the longest chord you can find and take one endpoint."
              ],
              "correctIndex": 1,
              "explanations": [
                "The centre is the midpoint of a chord only when that chord is a diameter — and finding a diameter is essentially the problem being solved.",
                "Correct. The perpendicular bisector of any chord passes through the centre, so two of them (from non-parallel chords) intersect at exactly one point: the centre.",
                "This assumes the radius is already known, which it is not — and the direction to measure would still be unknown.",
                "An endpoint of a chord lies on the circle, not at the centre. The longest chord IS a diameter, but its endpoints are as far from the centre as possible."
              ],
              "hint": "Which line related to a chord is guaranteed to pass through the centre?",
              "kind": "reasoning"
            }
          ]
        },
        {
          "id": "circles-circumference-area",
          "title": "Circumference, Area & Sectors",
          "blurb": "Circumference and area from a radius or a diameter, plus arc length and sector area.",
          "meta": "6 questions · fluency + reasoning",
          "categories": [
            "circles",
            "area-perimeter",
            "measurement"
          ],
          "questions": [
            {
              "id": "cg-009",
              "topic": "Circumference & Area",
              "difficulty": "easy",
              "stem": "A circle has radius 7 cm. What is its circumference, in terms of π?",
              "choices": [
                "7π cm",
                "14π cm",
                "49π cm",
                "28π cm"
              ],
              "correctIndex": 1,
              "explanations": [
                "7π uses C = πr, which is missing the factor of 2. Circumference is π times the DIAMETER, and the diameter is twice the radius.",
                "Correct. C = 2πr = 2π(7) = 14π cm.",
                "49π is the AREA (πr² = π·49). Area is measured in square units; circumference is a length.",
                "28π uses 2πd rather than 2πr — it doubles the diameter instead of the radius, giving twice the correct answer."
              ],
              "hint": "Circumference is π times the diameter. What is the diameter here?"
            },
            {
              "id": "cg-010",
              "topic": "Circumference & Area",
              "difficulty": "easy",
              "stem": "A circle has diameter 18 in. What is its area, in terms of π?",
              "choices": [
                "18π in²",
                "36π in²",
                "81π in²",
                "324π in²"
              ],
              "correctIndex": 2,
              "explanations": [
                "18π is π times the diameter, which is the circumference — a length, not an area.",
                "36π is 2π(18), doubling the diameter. That is not a formula for either quantity here.",
                "Correct. The radius is 18 ÷ 2 = 9, so A = πr² = π(9²) = 81π in².",
                "324π squares the DIAMETER instead of the radius. The formula needs r², and r is half of 18."
              ],
              "hint": "The area formula takes the radius, and you have been given the diameter."
            },
            {
              "id": "cg-011",
              "topic": "Circumference & Area",
              "difficulty": "medium",
              "stem": "In a circle of radius 10, an arc is intercepted by a central angle of 72°. What is the length of that arc, in terms of π?",
              "choices": [
                "2π",
                "4π",
                "20π",
                "72π"
              ],
              "correctIndex": 1,
              "explanations": [
                "2π halves the correct answer — check the fraction: 72/360 simplifies to 1/5, not 1/10.",
                "Correct. The arc is 72/360 = 1/5 of the circle, and the full circumference is 2π(10) = 20π. One fifth of 20π is 4π.",
                "20π is the ENTIRE circumference. The arc is only the fraction of it cut off by a 72° angle.",
                "72π multiplies by the angle instead of taking the angle as a fraction of 360°."
              ],
              "hint": "What fraction of the whole circle does 72° represent?"
            },
            {
              "id": "cg-012",
              "topic": "Circumference & Area",
              "difficulty": "hard",
              "stem": "A sector of a circle of radius 6 has a central angle of 120°. What is the area of the sector, in terms of π?",
              "choices": [
                "4π",
                "12π",
                "24π",
                "36π"
              ],
              "correctIndex": 1,
              "explanations": [
                "4π is 120/360 of the CIRCUMFERENCE (12π), not of the area. That fraction of the circumference gives the arc length, which is a length rather than an area.",
                "Correct. The sector is 120/360 = 1/3 of the circle, and the full area is π(6²) = 36π. One third of 36π is 12π.",
                "24π is two thirds of the circle's area — the fraction has been taken the wrong way round (240/360 rather than 120/360).",
                "36π is the area of the WHOLE circle. The sector is only the slice cut off by the 120° angle."
              ],
              "hint": "Take the same fraction of the area that the angle is of 360°."
            },
            {
              "id": "cg-046",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "easy",
              "stem": "A circle has diameter 20. A student computes the circumference as C = 2π(20) = 40π. What went wrong?",
              "choices": [
                "Nothing; 40π is correct.",
                "They used the diameter in place of the radius in C = 2πr. The circumference is 20π.",
                "They should have used C = πr².",
                "They should have divided by 2 at the end, giving 10π."
              ],
              "correctIndex": 1,
              "explanations": [
                "40π is twice the correct value, because the diameter was substituted where the radius belongs.",
                "Correct. C = 2πr needs the RADIUS, which is 10. Either compute 2π(10) or use C = πd = π(20) — both give 20π.",
                "πr² is the AREA formula, and it produces square units rather than a length.",
                "Dividing 40π by 2 does reach 20π, but by accident — the fix is to substitute the radius, not to patch the result."
              ],
              "hint": "Write down which quantity each formula expects before substituting.",
              "kind": "reasoning"
            },
            {
              "id": "cg-047",
              "topic": "Reasoning: What's Missing",
              "difficulty": "medium",
              "stem": "A student is asked for the AREA of a sector with central angle 60° in a circle of radius 12, and answers \"4π, because 60/360 of the circumference 24π is 4π.\" What is the mistake?",
              "choices": [
                "They found the arc LENGTH rather than the sector area; the area is 24π.",
                "The fraction should be 60/180.",
                "They should have used the diameter instead of the radius.",
                "Nothing is wrong; 4π is the area."
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. Taking that fraction of the circumference gives the arc length. For AREA, take the same fraction of the circle's area: ⅙ × π(12²) = ⅙ × 144π = 24π.",
                "60/360 is the right fraction — a sector's share of the circle is its angle over 360°.",
                "The radius was used correctly. The error was taking a fraction of the wrong total.",
                "4π is a length, not an area, and the units alone give it away."
              ],
              "hint": "The fraction was right. Check what it was a fraction OF.",
              "kind": "reasoning"
            }
          ]
        },
        {
          "id": "circles-equations",
          "title": "Equations of Circles",
          "blurb": "Standard form, reading a centre and radius off an equation, and completing the square.",
          "meta": "6 questions · fluency + reasoning",
          "categories": [
            "circles-coordinate-plane"
          ],
          "questions": [
            {
              "id": "cg-013",
              "topic": "Equations of Circles",
              "difficulty": "easy",
              "stem": "Write the equation of the circle with centre (3, −5) and radius 4.",
              "choices": [
                "(x − 3)² + (y + 5)² = 16",
                "(x + 3)² + (y − 5)² = 16",
                "(x − 3)² + (y + 5)² = 4",
                "(x − 3)² + (y − 5)² = 16"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. The form is (x − h)² + (y − k)² = r². With h = 3 and k = −5, y − (−5) becomes y + 5, and r² = 4² = 16.",
                "The signs of both coordinates are flipped. The formula SUBTRACTS the centre coordinates, so a centre of (3, −5) gives (x − 3) and (y + 5).",
                "The right-hand side is the radius rather than the radius squared. The equation needs r², so 4 becomes 16.",
                "(y − 5) would put the centre at y = +5. The centre's y-coordinate is −5, and subtracting a negative gives (y + 5)."
              ],
              "hint": "In (x − h)² + (y − k)² = r², the centre is (h, k) — watch what happens to the sign when k is negative."
            },
            {
              "id": "cg-014",
              "topic": "Equations of Circles",
              "difficulty": "easy",
              "stem": "A circle has the equation (x + 2)² + (y − 7)² = 49. What are its centre and radius?",
              "choices": [
                "Centre (2, −7), radius 7",
                "Centre (−2, 7), radius 7",
                "Centre (−2, 7), radius 49",
                "Centre (2, −7), radius 49"
              ],
              "correctIndex": 1,
              "explanations": [
                "The signs are reversed. (x + 2) means x − (−2), so h = −2, not 2.",
                "Correct. (x + 2) gives h = −2, (y − 7) gives k = 7, and r = √49 = 7.",
                "The centre is right, but 49 is r², not r. Take the square root: r = 7.",
                "Both the signs and the radius are off — read (x + 2) as x − (−2), and take the square root of 49."
              ]
            },
            {
              "id": "cg-015",
              "topic": "Equations of Circles",
              "difficulty": "hard",
              "stem": "Find the centre and radius of the circle x² + y² − 6x + 8y − 11 = 0.",
              "choices": [
                "Centre (3, −4), radius 6",
                "Centre (−3, 4), radius 6",
                "Centre (3, −4), radius 36",
                "Centre (6, −8), radius √11"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. Completing the square gives (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36, so the centre is (3, −4) and r = 6.",
                "The signs are inverted. Completing the square on x² − 6x gives (x − 3)², which puts the centre at x = +3.",
                "The centre is right, but 36 is r². The radius is √36 = 6.",
                "(6, −8) reads the coefficients straight off the equation without completing the square. Each centre coordinate is HALF the coefficient, with the sign flipped."
              ],
              "hint": "Group the x terms and the y terms, then add the square of half each coefficient to both sides."
            },
            {
              "id": "cg-016",
              "topic": "Equations of Circles",
              "difficulty": "hard",
              "stem": "The endpoints of a diameter of a circle are (1, 2) and (7, 10). What is the equation of the circle?",
              "choices": [
                "(x − 4)² + (y − 6)² = 25",
                "(x − 4)² + (y − 6)² = 100",
                "(x − 3)² + (y − 4)² = 25",
                "(x − 8)² + (y − 12)² = 25"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. The centre is the midpoint, (4, 6). The diameter has length √(6² + 8²) = 10, so the radius is 5 and r² = 25.",
                "100 uses the DIAMETER squared. The equation needs the radius squared, and the radius is half the diameter.",
                "(3, 4) is the difference between the endpoints, not their midpoint. The midpoint averages the coordinates.",
                "(8, 12) is the SUM of the coordinates. The midpoint is the sum divided by two."
              ],
              "hint": "The centre of a circle is the midpoint of any diameter, and the radius is half the diameter's length."
            },
            {
              "id": "cg-040",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "hard",
              "stem": "A student converts x² + y² + 10x − 4y + 13 = 0 and writes the centre as (5, −2). What is the actual centre?",
              "choices": [
                "(5, −2)",
                "(−5, 2)",
                "(−10, 4)",
                "(10, −4)"
              ],
              "correctIndex": 1,
              "explanations": [
                "(5, −2) has both signs backwards. Completing the square on x² + 10x gives (x + 5)², which places the centre at x = −5.",
                "Correct. x² + 10x becomes (x + 5)² and y² − 4y becomes (y − 2)², so the centre is (−5, 2). Each coordinate is half the coefficient with the sign reversed.",
                "(−10, 4) uses the coefficients directly without halving them.",
                "(10, −4) neither halves nor reverses the signs."
              ],
              "hint": "Halve each coefficient, then flip its sign. Both steps, in that order.",
              "kind": "reasoning"
            },
            {
              "id": "cg-048",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "medium",
              "stem": "A student writes the equation of the circle with centre (0, 4) and radius 6 as x² + (y − 4)² = 6. What is wrong?",
              "choices": [
                "The centre's sign is wrong; it should be (y + 4).",
                "The right-hand side must be r², so it should be 36.",
                "The x term should be (x − 0)², which is not the same as x².",
                "Nothing is wrong."
              ],
              "correctIndex": 1,
              "explanations": [
                "(y − 4) is correct for a centre with k = 4 — the formula subtracts the centre coordinate.",
                "Correct. The standard form is (x − h)² + (y − k)² = r². The radius is 6, so the right-hand side is 6² = 36, not 6.",
                "(x − 0)² is exactly x², so writing it either way is fine.",
                "Leaving 6 rather than 36 describes a circle of radius √6, which is a different circle."
              ],
              "hint": "Which side of the equation holds the radius, and in what form?",
              "kind": "reasoning"
            }
          ]
        },
        {
          "id": "circles-inscribed-angles",
          "title": "Inscribed Angles & Cyclic Quadrilaterals",
          "blurb": "Half the arc, Thales' theorem, and the supplementary opposite angles of a cyclic quadrilateral.",
          "meta": "9 questions · fluency + reasoning",
          "categories": [
            "circles"
          ],
          "questions": [
            {
              "id": "cg-017",
              "topic": "Inscribed Angles",
              "difficulty": "easy",
              "stem": "Inscribed angle ∠ABC measures 35° in circle O. What is the measure of its intercepted arc AC?",
              "choices": [
                "17.5°",
                "35°",
                "70°",
                "145°"
              ],
              "correctIndex": 2,
              "explanations": [
                "17.5° halves the angle. The halving goes the other way: the ANGLE is half the arc, so the arc is twice the angle.",
                "35° treats this like a central angle. The vertex here is on the circle, not at the centre, so the angle and arc are not equal.",
                "Correct. An inscribed angle is half its intercepted arc, so the arc is twice the angle: 2 × 35° = 70°.",
                "145° is 180° − 35°. Supplements come up with cyclic quadrilaterals, not with a single inscribed angle."
              ],
              "hint": "The vertex sits on the circle. Which is bigger — the inscribed angle, or the arc it intercepts?",
              "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"76\" r=\"54\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><path d=\"M 75.8 107 A 54 54 0 0 0 166.8 103\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"4.5\" stroke-linecap=\"round\" opacity=\"0.85\"/><line x1=\"106\" y1=\"23.8\" x2=\"75.8\" y2=\"107\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"106\" y1=\"23.8\" x2=\"166.8\" y2=\"103\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"75.8\" cy=\"107\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"106\" cy=\"23.8\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"166.8\" cy=\"103\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"106\" y=\"15.8\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"64.8\" y=\"111\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"177.8\" y=\"107\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"106\" y=\"43.8\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">35°</text><text x=\"120\" y=\"126\" fill=\"#7c3aed\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
              "diagramCaption": "Not drawn to scale"
            },
            {
              "id": "cg-018",
              "topic": "Inscribed Angles",
              "difficulty": "easy",
              "stem": "In circle O, arc RS measures 140°. Point T lies on the major arc. What is the measure of inscribed angle ∠RTS?",
              "choices": [
                "35°",
                "70°",
                "140°",
                "220°"
              ],
              "correctIndex": 1,
              "explanations": [
                "35° divides by four. The inscribed angle is half the arc, which requires one division by two, not two.",
                "Correct. An inscribed angle is half its intercepted arc: 140° ÷ 2 = 70°.",
                "140° is the arc itself. Only a CENTRAL angle equals its arc; an inscribed angle is half of it.",
                "220° is the major arc (360° − 140°). The angle intercepts arc RS, and in any case an inscribed angle is smaller than its arc, never larger."
              ]
            },
            {
              "id": "cg-019",
              "topic": "Inscribed Angles",
              "difficulty": "easy",
              "stem": "In circle O, segment AB is a diameter and C is a point on the circle, distinct from A and B. What is the measure of ∠ACB?",
              "choices": [
                "45°",
                "60°",
                "90°",
                "It depends on where C is"
              ],
              "correctIndex": 2,
              "explanations": [
                "45° would be correct only if the intercepted arc were 90°. A diameter cuts off a semicircle of 180°.",
                "60° would need an intercepted arc of 120°, which a diameter does not produce.",
                "Correct. AB is a diameter, so it intercepts a semicircle of 180°. The inscribed angle is half of that: 90°. This is Thales' theorem — any angle inscribed in a semicircle is right.",
                "It genuinely does not depend on C. Wherever C sits on the circle, the intercepted arc is the same semicircle, so the angle is always 90° — which is exactly what makes this theorem useful."
              ],
              "hint": "What arc does a diameter intercept, and what is half of it?",
              "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"78\" r=\"54\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"66\" y1=\"78\" x2=\"174\" y2=\"78\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"140.2\" y1=\"27.9\" x2=\"66\" y2=\"78\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"140.2\" y1=\"27.9\" x2=\"174\" y2=\"78\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"66\" cy=\"78\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"174\" cy=\"78\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"140.2\" cy=\"27.9\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"120\" cy=\"78\" r=\"3.2\" fill=\"#1f2024\"/><text x=\"55\" y=\"82\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"185\" y=\"82\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"140.2\" y=\"19.9\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"120\" y=\"93\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">O</text><text x=\"140.2\" y=\"47.9\" fill=\"#1f2024\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>"
            },
            {
              "id": "cg-020",
              "topic": "Inscribed Angles",
              "difficulty": "medium",
              "stem": "Quadrilateral ABCD is inscribed in a circle. If ∠A = 105°, what is the measure of ∠C?",
              "choices": [
                "52.5°",
                "75°",
                "105°",
                "255°"
              ],
              "correctIndex": 1,
              "explanations": [
                "52.5° halves the given angle. Halving relates an inscribed angle to its arc, not one opposite angle to another.",
                "Correct. Opposite angles of a cyclic quadrilateral are supplementary, so ∠C = 180° − 105° = 75°.",
                "105° would make the opposite angles equal. That is true of a parallelogram, but not of a quadrilateral inscribed in a circle.",
                "255° is 360° − 105°. The four angles of a quadrilateral total 360°, but OPPOSITE pairs total 180° each."
              ],
              "hint": "∠A and ∠C intercept arcs that together make the whole circle. What does that make the two angles sum to?"
            },
            {
              "id": "cg-036",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "hard",
              "stem": "An inscribed angle in a circle measures 50°. A student writes: \"The intercepted arc is also 50°, because the angle and the arc are the same.\" What went wrong, and what is the arc?",
              "choices": [
                "Nothing went wrong; the arc is 50°.",
                "They used the central-angle relationship on an inscribed angle. The arc is 100°.",
                "They forgot the arc is a major arc. The arc is 310°.",
                "They should have used supplements. The arc is 130°."
              ],
              "correctIndex": 1,
              "explanations": [
                "The arc is not 50°. \"Angle equals arc\" holds only when the vertex is at the CENTRE.",
                "Correct. Angle = arc is the central-angle relationship, and it needs the vertex at the centre. An inscribed angle has its vertex on the circle and measures half its arc, so the arc is 2 × 50° = 100°.",
                "Major and minor is not the issue — the student's error was applying the wrong relationship, and the intercepted arc here is the minor arc of 100°.",
                "Supplements relate opposite angles of a cyclic quadrilateral. There is no quadrilateral here, and the relationship is a doubling."
              ],
              "hint": "Ask first where the vertex is. That single fact decides which relationship applies.",
              "kind": "reasoning"
            },
            {
              "id": "cg-039",
              "topic": "Reasoning: Choose the Theorem",
              "difficulty": "medium",
              "stem": "In circle O, points A and B lie on the circle and C and D are two other points on the major arc AB. Which reason explains why ∠ACB ≅ ∠ADB?",
              "choices": [
                "They are vertical angles.",
                "Both are inscribed angles intercepting the same arc AB, so each is half of it.",
                "Both are central angles.",
                "ACBD is a parallelogram, so opposite angles are congruent."
              ],
              "correctIndex": 1,
              "explanations": [
                "Vertical angles are formed by two intersecting lines. These angles have different vertices.",
                "Correct. Each angle has its vertex on the circle and intercepts arc AB, so each measures half of arc AB — which makes them equal regardless of where C and D sit.",
                "A central angle has its vertex at O. Both of these have vertices on the circle.",
                "ACBD need not be a parallelogram, and opposite angles of a CYCLIC quadrilateral are supplementary rather than congruent."
              ],
              "hint": "Both angles are half the same thing.",
              "kind": "reasoning"
            },
            {
              "id": "cg-042",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "medium",
              "stem": "Arc PQ measures 96°. A student finds the inscribed angle by computing 96° × 2 = 192°. Without doing any arithmetic, how can you tell this is wrong?",
              "choices": [
                "An inscribed angle can never be larger than 180°, and it must be smaller than its arc.",
                "The arc should have been 360° − 96° first.",
                "Inscribed angles are always right angles.",
                "The student should have added 96° and 180°."
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. An inscribed angle is HALF its arc, so it is always smaller than the arc — and an angle of 192° is impossible in a triangle-like figure regardless. The answer is 48°.",
                "The minor arc is the intercepted one here, and switching to the major arc would not fix a doubling that should have been a halving.",
                "Only angles inscribed in a semicircle are right. That is a special case, not the general rule.",
                "Adding is not the relationship. The inscribed angle is half the arc."
              ],
              "hint": "You do not need the number — just ask whether the answer is even possible.",
              "kind": "reasoning"
            },
            {
              "id": "cg-043",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "medium",
              "stem": "Quadrilateral WXYZ is inscribed in a circle. A student claims ∠W and ∠X must be supplementary because \"opposite angles of a cyclic quadrilateral add to 180°.\" What is wrong?",
              "choices": [
                "The theorem is wrong; cyclic quadrilateral angles are congruent, not supplementary.",
                "The theorem is right, but ∠W and ∠X are ADJACENT. The supplementary pairs are ∠W with ∠Y, and ∠X with ∠Z.",
                "The theorem only applies to squares inscribed in circles.",
                "Nothing is wrong."
              ],
              "correctIndex": 1,
              "explanations": [
                "The theorem is correctly stated — opposite angles of a cyclic quadrilateral are indeed supplementary.",
                "Correct. In WXYZ the vertices are in order, so W and X are next to each other. Opposite pairs are W–Y and X–Z, and those are the ones that sum to 180°.",
                "It applies to every quadrilateral inscribed in a circle, not only to squares.",
                "The theorem was quoted correctly but applied to the wrong pair of angles."
              ],
              "hint": "Read the vertex order in the name of the quadrilateral. Which vertices are actually opposite?",
              "kind": "reasoning"
            },
            {
              "id": "cg-045",
              "topic": "Reasoning: Choose the Theorem",
              "difficulty": "medium",
              "stem": "A triangle is inscribed in a circle so that one of its sides is a diameter. What can be concluded about the triangle, and on what grounds?",
              "choices": [
                "It is equilateral, because all radii are congruent.",
                "It is a right triangle, because an angle inscribed in a semicircle intercepts a 180° arc and is therefore 90°.",
                "It is isosceles, because two sides are radii.",
                "Nothing can be concluded without knowing the radius."
              ],
              "correctIndex": 1,
              "explanations": [
                "Radii being congruent does not make the triangle's SIDES congruent — two of the triangle's sides are chords, not radii.",
                "Correct. This is Thales' theorem. The angle opposite the diameter is inscribed in a semicircle, intercepting an arc of 180°, so it measures half of that: 90°.",
                "The two sides meeting at the third vertex are chords, not radii, and they are generally different lengths.",
                "The conclusion holds for every circle, whatever the radius — which is precisely what makes the theorem useful."
              ],
              "hint": "What arc does the diameter cut off, and what is half of it?",
              "kind": "reasoning"
            }
          ]
        },
        {
          "id": "circles-secant-tangent-angles",
          "title": "Secant & Tangent Angles",
          "blurb": "Vertex inside, on, or outside the circle — and which of add, halve or subtract each one calls for.",
          "meta": "10 questions · fluency + reasoning",
          "categories": [
            "circles"
          ],
          "questions": [
            {
              "id": "cg-021",
              "topic": "Chord & Secant Angles",
              "difficulty": "medium",
              "stem": "Two chords intersect inside a circle. The two arcs they intercept measure 80° and 40°. What is the measure of the angle formed at the intersection?",
              "choices": [
                "20°",
                "40°",
                "60°",
                "120°"
              ],
              "correctIndex": 2,
              "explanations": [
                "20° is half the DIFFERENCE of the arcs. Halving a difference is the rule for a vertex OUTSIDE the circle; this vertex is inside.",
                "40° is one of the given arcs, not a combination of them.",
                "Correct. An angle formed by two chords meeting inside a circle is half the SUM of the two intercepted arcs: ½(80° + 40°) = 60°.",
                "120° is the sum of the arcs without halving. The formula is half the sum."
              ],
              "hint": "Vertex inside the circle: add the two arcs. Vertex outside: subtract them. Either way, halve the result.",
              "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"76\" r=\"54\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"73.2\" y1=\"49\" x2=\"166.8\" y2=\"103\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"161.4\" y1=\"41.3\" x2=\"89\" y2=\"120.2\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"73.2\" cy=\"49\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"161.4\" cy=\"41.3\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"166.8\" cy=\"103\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"89\" cy=\"120.2\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"63.2\" y=\"47\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"171.4\" y=\"39.3\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"176.8\" y=\"109\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"79\" y=\"128.2\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">D</text><text x=\"150\" y=\"44\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">80°</text><text x=\"90\" y=\"114\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">40°</text><text x=\"132\" y=\"88\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>"
            },
            {
              "id": "cg-022",
              "topic": "Chord & Secant Angles",
              "difficulty": "medium",
              "stem": "Two secants are drawn to a circle from an external point P. They intercept a far arc of 110° and a near arc of 40°. What is the measure of ∠P?",
              "choices": [
                "35°",
                "55°",
                "70°",
                "75°"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. An angle formed outside the circle is half the DIFFERENCE of the intercepted arcs: ½(110° − 40°) = 35°.",
                "55° is half of the far arc alone, ignoring the near arc entirely.",
                "70° is the difference of the arcs without halving.",
                "75° is half the SUM of the arcs. Adding is the rule for a vertex inside the circle; an external vertex takes the difference."
              ],
              "hint": "The vertex is outside the circle. Does that call for the sum of the arcs or the difference?",
              "diagram": "<svg viewBox=\"0 0 240 158\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"148\" cy=\"78\" r=\"48\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"24\" y1=\"78\" x2=\"184.8\" y2=\"47.1\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"24\" y1=\"78\" x2=\"184.8\" y2=\"108.9\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"24\" cy=\"78\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"105.6\" cy=\"55.5\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"105.6\" cy=\"100.5\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"184.8\" cy=\"47.1\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"184.8\" cy=\"108.9\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"14\" y=\"82\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">P</text><text x=\"188\" y=\"82\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">110°</text><text x=\"114\" y=\"82\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">40°</text><text x=\"40\" y=\"72\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
              "diagramCaption": "Not drawn to scale"
            },
            {
              "id": "cg-023",
              "topic": "Chord & Secant Angles",
              "difficulty": "hard",
              "stem": "Two secants meet at an external point, forming a 25° angle. The nearer intercepted arc measures 50°. What is the measure of the farther arc?",
              "choices": [
                "25°",
                "75°",
                "100°",
                "125°"
              ],
              "correctIndex": 2,
              "explanations": [
                "25° is the angle itself, not an arc.",
                "75° comes from adding the angle to the near arc. The relationship involves half the difference, so the angle must be doubled before comparing arcs.",
                "Correct. 25° = ½(far − 50°), so far − 50° = 50°, giving far = 100°.",
                "125° adds the doubled angle to the near arc but then adds the angle once more. Double 25° to get 50°, then add the near arc: 50° + 50° = 100°."
              ],
              "hint": "Write the relationship as an equation and solve for the unknown arc rather than trying to do it in one step."
            },
            {
              "id": "cg-024",
              "topic": "Chord & Secant Angles",
              "difficulty": "medium",
              "stem": "Two chords intersect inside a circle, forming a 70° angle. One of the intercepted arcs measures 90°. What is the measure of the other intercepted arc?",
              "choices": [
                "20°",
                "50°",
                "80°",
                "140°"
              ],
              "correctIndex": 1,
              "explanations": [
                "20° comes from 90° − 70°, subtracting the angle directly from the arc. The angle is half the SUM of the arcs, so it must be doubled first.",
                "Correct. 70° = ½(90° + x), so 140° = 90° + x, giving x = 50°.",
                "80° does not satisfy the relationship: ½(90° + 80°) = 85°, not 70°.",
                "140° is twice the angle, which equals the SUM of both arcs — the second arc is what remains after subtracting the first."
              ],
              "hint": "Double the angle first. That gives you the sum of the two arcs."
            },
            {
              "id": "cg-025",
              "topic": "Tangent Angles",
              "difficulty": "medium",
              "stem": "A tangent and a chord meet at a point on a circle. The chord cuts off an arc of 130° on the side of the angle. What is the measure of the angle between the tangent and the chord?",
              "choices": [
                "32.5°",
                "65°",
                "115°",
                "130°"
              ],
              "correctIndex": 1,
              "explanations": [
                "32.5° divides by four. The tangent–chord angle is half the arc, which is a single division by two.",
                "Correct. A tangent–chord angle equals half its intercepted arc: ½(130°) = 65°.",
                "115° is 180° − 65°, the angle on the OTHER side of the tangent line. That angle intercepts the other arc.",
                "130° is the arc itself. Only a central angle equals its arc."
              ],
              "hint": "A tangent–chord angle follows the same half-the-arc rule as an inscribed angle — think of the tangent as a chord whose second endpoint has slid into T.",
              "diagram": "<svg viewBox=\"0 0 240 148\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"66\" r=\"50\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><path d=\"M 167 48.9 A 50 50 0 1 0 120 116\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"4.5\" stroke-linecap=\"round\" opacity=\"0.85\"/><line x1=\"40\" y1=\"116\" x2=\"200\" y2=\"116\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"116\" x2=\"167\" y2=\"48.9\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"116\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"167\" cy=\"48.9\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"120\" y=\"133\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">T</text><text x=\"178\" y=\"50.9\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"142\" y=\"108\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"160\" y=\"92\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">130°</text></svg>",
              "diagramCaption": "The horizontal line is tangent at T"
            },
            {
              "id": "cg-026",
              "topic": "Tangent Angles",
              "difficulty": "medium",
              "stem": "From an external point, a secant and a tangent are drawn to a circle. They intercept a far arc of 150° and a near arc of 60°. What is the measure of the angle at the external point?",
              "choices": [
                "45°",
                "75°",
                "90°",
                "105°"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. Any angle with its vertex outside the circle is half the difference of the intercepted arcs: ½(150° − 60°) = 45°.",
                "75° is half the far arc alone. Both arcs matter.",
                "90° is the difference of the arcs without halving.",
                "105° is half the SUM. Summing applies to a vertex inside the circle; this vertex is outside."
              ]
            },
            {
              "id": "cg-027",
              "topic": "Tangent Angles",
              "difficulty": "medium",
              "stem": "Two tangents are drawn to a circle from the same external point. They intercept a major arc of 210° and a minor arc of 150°. What is the measure of the angle between the tangents?",
              "choices": [
                "30°",
                "60°",
                "90°",
                "180°"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. The vertex is outside the circle, so the angle is half the difference: ½(210° − 150°) = 30°.",
                "60° is the difference of the arcs without halving.",
                "90° would require the arcs to differ by 180°. They differ by 60°.",
                "180° is the sum of the arcs halved (½ × 360°), which would mean the two tangents form a straight line — impossible for two tangents meeting at a point."
              ],
              "hint": "Two tangents from one point still make an external angle, so the same half-the-difference rule applies."
            },
            {
              "id": "cg-028",
              "topic": "Tangent Angles",
              "difficulty": "hard",
              "stem": "Two tangents from an external point form a 40° angle. What is the measure of the minor arc between the two points of tangency?",
              "choices": [
                "40°",
                "80°",
                "140°",
                "220°"
              ],
              "correctIndex": 2,
              "explanations": [
                "40° is the angle itself. The angle and the minor arc are not equal for an external vertex.",
                "80° is twice the angle, which gives the DIFFERENCE between the two arcs — not the minor arc itself.",
                "Correct. The arcs sum to 360° and differ by 2 × 40° = 80°. Solving gives major = 220° and minor = 140°.",
                "220° is the MAJOR arc. The question asks for the minor one."
              ],
              "hint": "Set up two facts about the arcs: they add to 360°, and their difference is twice the angle."
            },
            {
              "id": "cg-037",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "hard",
              "stem": "Two secants meet outside a circle, intercepting arcs of 120° and 40°. A student computes the angle as ½(120° + 40°) = 80°. What is the error?",
              "choices": [
                "The arcs should be added but not halved; the angle is 160°.",
                "An external vertex takes half the DIFFERENCE, not half the sum; the angle is 40°.",
                "The student should have used only the far arc; the angle is 60°.",
                "There is no error; the angle is 80°."
              ],
              "correctIndex": 1,
              "explanations": [
                "Halving is correct — it is the addition that is wrong. External angles use the difference.",
                "Correct. Adding the arcs is the rule for a vertex INSIDE the circle. Outside the circle the angle is half the difference: ½(120° − 40°) = 40°.",
                "Both arcs are needed. Using only the far arc would give 60°, which ignores how far outside the circle the vertex sits.",
                "80° is what the inside-the-circle rule gives. The vertex here is outside, so the rule does not apply."
              ],
              "hint": "Inside the circle: add. Outside the circle: subtract. Which is this?",
              "kind": "reasoning"
            },
            {
              "id": "cg-050",
              "topic": "Reasoning: Choose the Theorem",
              "difficulty": "hard",
              "stem": "In circle O, a tangent meets a chord at point T, forming a 40° angle. Separately, an inscribed angle elsewhere on the circle intercepts the same arc. What is that inscribed angle, and why?",
              "choices": [
                "20°, because an inscribed angle is half a tangent–chord angle.",
                "40°, because both a tangent–chord angle and an inscribed angle equal half their intercepted arc.",
                "80°, because the inscribed angle doubles the tangent–chord angle.",
                "50°, because they are complementary."
              ],
              "correctIndex": 1,
              "explanations": [
                "There is no halving between the two — both are already half of the same arc, so they are equal.",
                "Correct. A tangent–chord angle is half its intercepted arc, and so is an inscribed angle intercepting that same arc. Both equal ½(80°) = 40°.",
                "Doubling the tangent–chord angle gives the ARC (80°), not the inscribed angle.",
                "Complementary angles sum to 90°, which has no bearing on these two."
              ],
              "hint": "Work out the arc from the tangent–chord angle first, then use it for the inscribed angle.",
              "kind": "reasoning"
            }
          ]
        },
        {
          "id": "circles-segments-tangents",
          "title": "Segment Lengths & Tangents",
          "blurb": "Power of a point for chords, secants and tangents, and the right angle a tangent makes with a radius.",
          "meta": "9 questions · fluency + reasoning",
          "categories": [
            "circles"
          ],
          "questions": [
            {
              "id": "cg-029",
              "topic": "Segment Lengths",
              "difficulty": "medium",
              "stem": "Two chords intersect inside a circle. One is divided into segments of length 6 and 8; the other into segments of length 4 and x. What is x?",
              "choices": [
                "2",
                "10",
                "12",
                "48"
              ],
              "correctIndex": 2,
              "explanations": [
                "2 comes from dividing 8 by 4. The products of the two pieces of each chord are equal, so set 6 × 8 equal to 4 × x.",
                "10 comes from 6 + 8 − 4, combining the lengths by addition. The relationship is multiplicative.",
                "Correct. The two chords satisfy 6 × 8 = 4 × x, so 48 = 4x and x = 12.",
                "48 is the product 6 × 8, which equals 4x — not x itself. Divide by 4 to finish."
              ],
              "hint": "For two chords crossing inside a circle, the product of one chord's two pieces equals the product of the other's.",
              "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"76\" r=\"54\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"75.8\" y1=\"45\" x2=\"164.2\" y2=\"107\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"151\" y1=\"31.8\" x2=\"93\" y2=\"122.8\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"76\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"75.8\" cy=\"45\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"151\" cy=\"31.8\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"164.2\" cy=\"107\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"93\" cy=\"122.8\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"97.9\" y=\"54.5\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">6</text><text x=\"142.1\" y=\"105.5\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">8</text><text x=\"147.5\" y=\"53.9\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4</text><text x=\"94.5\" y=\"99.4\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text></svg>"
            },
            {
              "id": "cg-030",
              "topic": "Segment Lengths",
              "difficulty": "hard",
              "stem": "From external point P, one secant has external segment 4 and total length 15. A second secant from P has external segment 5. What is the total length of the second secant?",
              "choices": [
                "7",
                "12",
                "16",
                "18.75"
              ],
              "correctIndex": 1,
              "explanations": [
                "7 is the far portion of the second secant (12 − 5), not its total length. The relationship uses the WHOLE secant.",
                "Correct. For two secants from the same point, (external)(whole) is equal for both: 4 × 15 = 5 × whole, so whole = 60 ÷ 5 = 12.",
                "16 does not satisfy 5 × whole = 60.",
                "18.75 divides 75 by 4, mixing up which numbers pair together. The known secant contributes 4 × 15 = 60."
              ],
              "hint": "Each secant contributes (external part) × (whole secant), and the two products are equal."
            },
            {
              "id": "cg-031",
              "topic": "Segment Lengths",
              "difficulty": "hard",
              "stem": "A tangent of length 12 and a secant are drawn to a circle from the same external point. The secant's external segment measures 8. What is the length of the secant's internal chord?",
              "choices": [
                "10",
                "18",
                "1.5",
                "4"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. The tangent-secant relationship gives 12² = 8 × (whole), so the whole secant is 144 ÷ 8 = 18. The internal chord is 18 − 8 = 10.",
                "18 is the WHOLE secant. The internal chord is what remains after removing the external segment of 8.",
                "1.5 divides 12 by 8 without squaring the tangent. The tangent length appears squared in this relationship.",
                "4 is 12 − 8, subtracting the lengths directly. The relationship is multiplicative, not additive."
              ],
              "hint": "The tangent squared equals the external segment times the WHOLE secant — then read carefully which piece is asked for."
            },
            {
              "id": "cg-032",
              "topic": "Segment Lengths",
              "difficulty": "medium",
              "stem": "Two chords intersect inside a circle. One is split into segments of 9 and x; the other into segments of 6 and 12. What is x?",
              "choices": [
                "3",
                "8",
                "15",
                "72"
              ],
              "correctIndex": 1,
              "explanations": [
                "3 comes from 9 − 6, subtracting the pieces. The relationship multiplies them.",
                "Correct. 9x = 6 × 12 = 72, so x = 8.",
                "15 comes from adding 6 and 12 and subtracting 3. The two chords relate by equal products, not by sums.",
                "72 is the product 6 × 12, which equals 9x — divide by 9 to get x."
              ]
            },
            {
              "id": "cg-033",
              "topic": "Tangents",
              "difficulty": "medium",
              "stem": "PA is tangent to circle O at point A. The radius OA measures 9 and the tangent PA measures 12. What is the distance OP?",
              "choices": [
                "15",
                "21",
                "√63",
                "3"
              ],
              "correctIndex": 0,
              "explanations": [
                "Correct. A tangent is perpendicular to the radius at the point of tangency, so OAP is a right triangle with legs 9 and 12. OP = √(81 + 144) = √225 = 15.",
                "21 adds the two lengths. They are the legs of a right triangle, so they combine by the Pythagorean theorem instead.",
                "√63 comes from 144 − 81, subtracting the squares. That would find a LEG; here the unknown is the hypotenuse, so the squares are added.",
                "3 is 12 − 9. The three lengths form a right triangle, not a straight line."
              ],
              "hint": "What is the angle between a tangent and the radius drawn to its point of tangency?",
              "diagram": "<svg viewBox=\"0 0 240 152\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"92\" cy=\"84\" r=\"42\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"92\" y1=\"84\" x2=\"113\" y2=\"47.6\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"113\" y1=\"47.6\" x2=\"196\" y2=\"44\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"92\" y1=\"84\" x2=\"196\" y2=\"44\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"92\" cy=\"84\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"113\" cy=\"47.6\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"196\" cy=\"44\" r=\"3.2\" fill=\"#1f2024\"/><text x=\"82\" y=\"88\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">O</text><text x=\"107\" y=\"39.6\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"206\" y=\"46\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">P</text><text x=\"92.5\" y=\"65.8\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">9</text><text x=\"154.5\" y=\"39.8\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">12</text><text x=\"144\" y=\"80\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" text-anchor=\"middle\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
              "diagramCaption": "A tangent meets the radius at its point of tangency"
            },
            {
              "id": "cg-034",
              "topic": "Tangents",
              "difficulty": "medium",
              "stem": "Two tangent segments are drawn to a circle from external point Q. One measures 3x − 2 and the other measures x + 8. What is the value of x?",
              "choices": [
                "3",
                "5",
                "6",
                "10"
              ],
              "correctIndex": 1,
              "explanations": [
                "3 gives tangent lengths of 7 and 11, which are not equal.",
                "Correct. Two tangent segments from the same external point are congruent, so 3x − 2 = x + 8. Then 2x = 10 and x = 5, making both tangents 13.",
                "6 gives lengths of 16 and 14 — close, but not equal.",
                "10 gives 28 and 18, which are not equal. Set the two expressions equal to each other and solve."
              ],
              "hint": "What is always true about two tangent segments drawn to a circle from the same outside point?"
            },
            {
              "id": "cg-035",
              "topic": "Tangents",
              "difficulty": "medium",
              "stem": "PT is tangent to circle O at T. If the radius of the circle is 8 and OP = 17, what is the length of the tangent segment PT?",
              "choices": [
                "9",
                "15",
                "√353",
                "25"
              ],
              "correctIndex": 1,
              "explanations": [
                "9 is 17 − 8, subtracting the lengths directly. They are sides of a right triangle, not points on a line.",
                "Correct. The tangent is perpendicular to the radius, so PT = √(17² − 8²) = √(289 − 64) = √225 = 15.",
                "√353 comes from ADDING the squares. Here the hypotenuse (OP = 17) is known and a leg is unknown, so the squares are subtracted.",
                "25 adds the two given lengths. The Pythagorean relationship applies instead."
              ]
            },
            {
              "id": "cg-038",
              "topic": "Reasoning: Choose the Theorem",
              "difficulty": "medium",
              "stem": "Two tangent segments are drawn to a circle from the same external point. Which statement justifies concluding that the two segments are congruent?",
              "choices": [
                "Inscribed angles that intercept the same arc are congruent.",
                "The two right triangles formed with the radii are congruent by Hypotenuse-Leg.",
                "The two tangents intercept congruent arcs.",
                "Chords equidistant from the centre are congruent."
              ],
              "correctIndex": 1,
              "explanations": [
                "Inscribed angles are not involved — nothing here has its vertex on the circle intercepting a shared arc.",
                "Correct. Draw radii to both points of tangency. Each radius is perpendicular to its tangent, the radii are congruent, and the two triangles share the hypotenuse from the centre to the external point — so they are congruent by HL, making the tangent segments congruent.",
                "The two tangents intercept arcs of different sizes (a major and a minor arc), so this is not true, and it would not prove the segments congruent even if it were.",
                "This is a real theorem, but it is about chords. Tangent segments lie outside the circle and are not chords."
              ],
              "hint": "What can you draw from the centre to each point of tangency, and what does that create?",
              "kind": "reasoning"
            },
            {
              "id": "cg-049",
              "topic": "Reasoning: Error Analysis",
              "difficulty": "hard",
              "stem": "Two chords cross inside a circle, splitting one chord into 5 and 9 and the other into 3 and y. A student writes 5 + 9 = 3 + y and gets y = 11. What is the correct relationship and value?",
              "choices": [
                "The sums are equal; y = 11 is correct.",
                "The PRODUCTS are equal: 5 × 9 = 3 × y, so y = 15.",
                "The differences are equal: 9 − 5 = y − 3, so y = 7.",
                "The chords must be congruent, so y = 9."
              ],
              "correctIndex": 1,
              "explanations": [
                "Equal sums would mean the two chords are the same length, which crossing chords need not be.",
                "Correct. When two chords intersect inside a circle, the products of the two pieces are equal: 5 × 9 = 45, so 3y = 45 and y = 15.",
                "Differences are not preserved. The relationship comes from similar triangles, which produce equal ratios and therefore equal products.",
                "Two chords crossing inside a circle are generally not congruent, and nothing here says these are."
              ],
              "hint": "The relationship comes from a pair of similar triangles, so it produces a proportion — and a proportion cross-multiplies.",
              "kind": "reasoning"
            }
          ]
        }
      ]
    }
  }
}