{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Circle Theorems (Advanced)",
    "author": "Staples Education",
    "description": "Inscribed angles and arcs, intersecting chords, secants and tangents from an exterior point, the power-of-a-point theorems, cyclic quadrilaterals, chord geometry, sector area, and circles on the coordinate plane. Converted from worked tutoring whiteboards on circle theorems.",
    "created": "2026-08-06T04:14:56.202Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "circle-theorems-advanced",
      "title": "Circle Theorems (Advanced)",
      "blurb": "Inscribed angles, chords crossing inside and secants meeting outside, the tangent-secant power theorem, cyclic quadrilaterals, and sector area.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "circles"
      ],
      "questions": [
        {
          "id": "cta-001",
          "topic": "Inscribed Angles",
          "difficulty": "easy",
          "stem": "An inscribed angle intercepts an arc measuring 110°. What is the measure of the inscribed angle?",
          "choices": [
            "220°",
            "55°",
            "70°",
            "110°"
          ],
          "correctIndex": 1,
          "explanations": [
            "220° doubles the arc instead of halving it. That relationship runs the other way: the ARC is twice the inscribed angle, so 110° would be the angle and 220° the arc.",
            "Correct. An inscribed angle is always half the measure of its intercepted arc: ½(110°) = 55°.",
            "70° does not follow from any relationship here. It might come from 180° − 110°, which is the supplement — a fact about cyclic quadrilaterals, not about a single inscribed angle.",
            "110° repeats the arc. That would be the measure of the CENTRAL angle standing on the same arc; an inscribed angle on the same arc is half of it."
          ],
          "hint": "Central angle equals its arc. An inscribed angle on the same arc is half that.",
          "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=\"70\" r=\"55\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"120\" y1=\"15\" x2=\"78\" y2=\"105\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"15\" x2=\"165\" y2=\"100\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"15\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"78\" cy=\"105\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"165\" cy=\"100\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"112\" y=\"34\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"94\" y=\"138\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">arc = 110°</text></svg>",
          "diagramCaption": "The inscribed angle stands on the marked arc. Not drawn to scale"
        },
        {
          "id": "cta-002",
          "topic": "Intersecting Chords",
          "difficulty": "medium",
          "stem": "Two chords intersect inside a circle. The intercepted arcs are 80° and 40°. What is the measure of the vertical angles formed at their intersection?",
          "choices": [
            "20°",
            "120°",
            "40°",
            "60°"
          ],
          "correctIndex": 3,
          "explanations": [
            "20° halves the DIFFERENCE of the arcs, ½(80 − 40). The difference rule applies when the vertex is OUTSIDE the circle. Inside, the arcs add.",
            "120° is the sum of the two arcs without halving. The angle is half that sum, not the sum itself.",
            "40° repeats one of the given arcs. Both arcs contribute to the angle.",
            "Correct. When two chords intersect inside a circle, the angle is half the SUM of the two intercepted arcs: ½(80° + 40°) = ½(120°) = 60°."
          ],
          "hint": "Vertex inside the circle or outside? That decides whether the two arcs are added or subtracted before halving.",
          "diagram": "<svg viewBox=\"0 0 240 145\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"72\" r=\"58\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"70\" y1=\"42\" x2=\"178\" y2=\"96\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"74\" y1=\"105\" x2=\"172\" y2=\"40\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"122\" cy=\"72\" r=\"3.2\" fill=\"#c0392b\"/><text x=\"108\" y=\"24\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">80°</text><text x=\"108\" y=\"128\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">40°</text><text x=\"128\" y=\"68\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
          "diagramCaption": "Two chords crossing inside the circle. Not drawn to scale"
        },
        {
          "id": "cta-003",
          "topic": "Secants and Tangents",
          "difficulty": "medium",
          "stem": "Two secants intersect at a point OUTSIDE a circle. The intercepted arcs measure 120° and 30°. What is the angle formed at the intersection point?",
          "choices": [
            "90°",
            "75°",
            "15°",
            "45°"
          ],
          "correctIndex": 3,
          "explanations": [
            "90° is the difference 120° − 30° without halving. The halving step is not optional.",
            "75° halves the SUM, ½(120 + 30). The sum rule applies when the vertex is inside the circle; outside, the arcs subtract.",
            "15° halves 30°, using only the near arc. Both arcs matter — the far one carries the positive contribution.",
            "Correct. When the vertex is outside the circle, the angle is half the DIFFERENCE of the intercepted arcs: ½(120° − 30°) = ½(90°) = 45°. The far arc is the larger one, and the near arc is subtracted from it."
          ],
          "hint": "The vertex has moved outside the circle. That flips the sum in the intersecting-chords rule to a difference.",
          "diagram": "<svg viewBox=\"0 0 240 140\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"150\" cy=\"70\" r=\"52\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"20\" y1=\"70\" x2=\"200\" y2=\"40\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"20\" y1=\"70\" x2=\"200\" y2=\"104\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"20\" cy=\"70\" r=\"3.2\" fill=\"#c0392b\"/><text x=\"26\" y=\"62\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"160\" y=\"26\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">120°</text><text x=\"102\" y=\"82\" fill=\"#7c3aed\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">30°</text></svg>",
          "diagramCaption": "Both secants meet outside the circle. Not drawn to scale"
        },
        {
          "id": "cta-004",
          "topic": "Power of a Point",
          "difficulty": "medium",
          "stem": "A tangent segment of length t and a secant segment are drawn to a circle from an exterior point. The external segment of the secant is 4 and the whole secant is 16. What is t?",
          "choices": [
            "64",
            "8",
            "12",
            "10"
          ],
          "correctIndex": 1,
          "explanations": [
            "64 is t², not t. The Tangent-Secant Power Theorem gives t² = 64, and the square root still has to be taken.",
            "Correct. The Tangent-Secant Power Theorem states t² = (whole secant) × (external segment) = 16 × 4 = 64, so t = √64 = 8.",
            "12 would be the internal part of the secant, 16 − 4. That length appears in the figure but is not what the theorem multiplies — the theorem uses the WHOLE secant and its external part.",
            "10 is the average of 4 and 16. The relationship is a geometric mean, not an arithmetic one: t = √(4 × 16), not (4 + 16)/2."
          ],
          "hint": "Square the tangent, and multiply the whole secant by its outside piece. Then remember which side of the equation you were asked for.",
          "diagram": "<svg viewBox=\"0 0 240 142\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"155\" cy=\"72\" r=\"48\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"22\" y1=\"96\" x2=\"122\" y2=\"38\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"22\" y1=\"96\" x2=\"210\" y2=\"82\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"22\" cy=\"96\" r=\"3.2\" fill=\"#c0392b\"/><circle cx=\"122\" cy=\"38\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"63\" cy=\"92\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"203\" cy=\"78\" r=\"3.2\" fill=\"#7c3aed\"/><path d=\"M 116,44 L 124,52 L 132,44\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"56\" y=\"56\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">t</text><text x=\"30\" y=\"112\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4</text><text x=\"120\" y=\"108\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">16 (whole)</text></svg>",
          "diagramCaption": "Tangent and secant from one exterior point. Not drawn to scale"
        },
        {
          "id": "cta-005",
          "topic": "Inscribed Quadrilaterals",
          "difficulty": "medium",
          "stem": "Quadrilateral JKML is inscribed in a circle. Inscribed angle ∠K measures 135°. What is the measure of the major arc JLM that ∠K intercepts?",
          "choices": [
            "67.5°",
            "270°",
            "225°",
            "135°"
          ],
          "correctIndex": 1,
          "explanations": [
            "67.5° halves the angle instead of doubling it. Halving goes from arc to inscribed angle; here you are travelling the other way.",
            "Correct. The inscribed angle is half its intercepted arc, so the arc is twice the angle: 2(135°) = 270°. ∠K is the angle that intercepts major arc JLM, since the arc it opens onto is the one NOT containing its own vertex. The remaining arc is then 360° − 270° = 90°.",
            "225° does not follow from doubling or halving 135°. Check the arithmetic: 2 × 135 = 270.",
            "135° repeats the angle. An inscribed angle and its intercepted arc are equal only when both are zero — the arc is always twice the angle."
          ],
          "hint": "Which arc does an inscribed angle open onto — the one containing its vertex, or the other one? Then double.",
          "diagram": "<svg viewBox=\"0 0 240 146\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"74\" r=\"58\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><polygon points=\"80,30 165,52 168,104 78,116\" fill=\"none\" stroke=\"#1f2024\" stroke-width=\"1.8\" stroke-linejoin=\"round\"/><circle cx=\"80\" cy=\"30\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"165\" cy=\"52\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"168\" cy=\"104\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"78\" cy=\"116\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"64\" y=\"26\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">J</text><text x=\"170\" y=\"46\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">K</text><text x=\"172\" y=\"116\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">M</text><text x=\"62\" y=\"126\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">L</text><text x=\"140\" y=\"66\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">135°</text></svg>",
          "diagramCaption": "JKML is a cyclic quadrilateral. Not drawn to scale"
        },
        {
          "id": "cta-006",
          "topic": "Inscribed Quadrilaterals",
          "difficulty": "easy",
          "stem": "A quadrilateral is inscribed in a circle. One of its angles measures 72°. What is the measure of the angle opposite it?",
          "choices": [
            "72°",
            "108°",
            "18°",
            "288°"
          ],
          "correctIndex": 1,
          "explanations": [
            "72° would make the opposite angles congruent, which is a parallelogram property, not a cyclic one. Opposite angles of an inscribed quadrilateral are supplementary instead.",
            "Correct. The Inscribed Quadrilateral Theorem says opposite angles of a cyclic quadrilateral are supplementary: 180° − 72° = 108°. The reason is that the two angles between them intercept arcs making up the whole circle, so their halves total half of 360°.",
            "18° is the complement, 90° − 72°. Complementary angles sum to 90°; these sum to 180°.",
            "288° is 360° − 72°, the reflex arc rather than an angle of the quadrilateral. No interior angle of a convex quadrilateral exceeds 180°."
          ],
          "hint": "The two arcs the opposite angles stand on together make up the entire circle. Halve 360° and see what that forces."
        },
        {
          "id": "cta-007",
          "topic": "Chords",
          "difficulty": "medium",
          "stem": "Two chords of a circle are congruent. One is labelled 6x − 7 and the other measures 35. What is the value of x?",
          "choices": [
            "7",
            "5",
            "42",
            "17.5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Congruent chords have equal length, so 6x − 7 = 35. Adding 7 gives 6x = 42, and dividing by 6 gives x = 7.",
            "5 would follow from 6x = 30, which drops the −7 or subtracts it on the wrong side. The constant has to be added to 35, not taken from it.",
            "42 is 6x, one step short. It still has to be divided by 6.",
            "17.5 is half of 35 — the half-chord you would use when dropping a perpendicular from the centre. That is a later step in a different problem, not the value of x."
          ],
          "hint": "Congruent means equal in length. Set the two expressions equal and solve — nothing circular is needed for this step."
        },
        {
          "id": "cta-008",
          "topic": "Chords",
          "difficulty": "hard",
          "stem": "A chord of length 35 lies in a circle whose centre is 12 units from the chord. What is the radius of the circle, to two decimal places?",
          "choices": [
            "21.22",
            "37.00",
            "18.50",
            "24.04"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A perpendicular dropped from the centre to a chord bisects it, giving a half-chord of 35/2 = 17.5. That half-chord, the distance 12, and the radius form a right triangle: 12² + 17.5² = r², so 144 + 306.25 = 450.25 and r = √450.25 ≈ 21.22.",
            "37.00 is 12 + 35 · … no combination of the given numbers by addition gives the radius. The relationship is Pythagorean, so the two legs must be squared before they are combined.",
            "18.50 is the half-chord 17.5 nudged, or half of 37. The half-chord is one leg of the right triangle, not the hypotenuse — the radius is longer than either leg.",
            "24.04 is √(12² + 21²) or a similar slip using the full chord somewhere it should be halved. Only HALF the chord sits in the right triangle."
          ],
          "hint": "Drop a perpendicular from the centre to the chord. It bisects the chord — so only half of 35 belongs to the right triangle you are about to solve.",
          "diagram": "<svg viewBox=\"0 0 240 160\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"72\" r=\"62\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"58\" y1=\"100\" x2=\"182\" y2=\"100\" stroke=\"#1f2024\" stroke-width=\"2.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"72\" x2=\"120\" y2=\"100\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"72\" x2=\"182\" y2=\"100\" stroke=\"#c0392b\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><path d=\"M 120,90 L 132,90 L 132,100\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><circle cx=\"120\" cy=\"72\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"126\" y=\"88\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">12</text><text x=\"140\" y=\"114\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">17.5</text><text x=\"146\" y=\"78\" fill=\"#c0392b\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">r</text><text x=\"96\" y=\"150\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">chord = 35</text></svg>",
          "diagramCaption": "The perpendicular from the centre bisects the chord. Not drawn to scale"
        },
        {
          "id": "cta-009",
          "topic": "Secants and Tangents",
          "difficulty": "medium",
          "stem": "From an exterior point H, a tangent touches a circle at G. The radius to G measures 5 cm and HG measures 8 cm. How far is H from the centre J?",
          "choices": [
            "13 cm",
            "√89 cm",
            "√39 cm",
            "40 cm"
          ],
          "correctIndex": 1,
          "explanations": [
            "13 cm adds the two lengths, 5 + 8. They are the legs of a right triangle, not collinear segments, so they combine through the Pythagorean theorem rather than by addition.",
            "Correct. By the Tangent-Radius Theorem the radius JG is perpendicular to the tangent at G, so triangle JGH has a right angle at G. Then HJ² = 8² + 5² = 64 + 25 = 89, giving HJ = √89 cm (about 9.43 cm).",
            "√39 comes from 64 − 25, subtracting the legs instead of adding them. Subtraction is right when solving for a LEG given the hypotenuse; here the hypotenuse is the unknown.",
            "40 is 5 × 8. Multiplying the legs gives twice the triangle's area, not a length."
          ],
          "hint": "What angle does a tangent make with the radius drawn to the point of tangency? That turns the figure into a right triangle.",
          "diagram": "<svg viewBox=\"0 0 240 130\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"150\" cy=\"70\" r=\"42\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"25\" y1=\"88\" x2=\"128\" y2=\"40\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"128\" y1=\"40\" x2=\"150\" y2=\"70\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"88\" x2=\"150\" y2=\"70\" stroke=\"#c0392b\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><path d=\"M 122,48 L 132,54 L 126,62\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><circle cx=\"128\" cy=\"40\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"150\" cy=\"70\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"25\" cy=\"88\" r=\"3.2\" fill=\"#c0392b\"/><text x=\"14\" y=\"96\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">H</text><text x=\"122\" y=\"32\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">G</text><text x=\"156\" y=\"66\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">J</text><text x=\"60\" y=\"56\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">8 cm</text><text x=\"140\" y=\"60\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5</text><text x=\"74\" y=\"92\" fill=\"#c0392b\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
          "diagramCaption": "HG is tangent at G, so JG ⟂ HG. Not drawn to scale"
        },
        {
          "id": "cta-010",
          "topic": "Secants and Tangents",
          "difficulty": "hard",
          "stem": "Two tangent segments HG and HI are drawn from an exterior point H to a circle centred at J. If ∠GHJ = θ, what is m∠GHI in terms of θ, and why?",
          "choices": [
            "θ, because the tangents are congruent",
            "2θ, because HJ bisects ∠GHI",
            "90° − θ, because the radius is perpendicular to the tangent",
            "180° − θ, because GHIJ is cyclic"
          ],
          "correctIndex": 1,
          "explanations": [
            "The tangent segments ARE congruent by the Two Tangents Theorem, but that is what makes HJ a line of symmetry — it does not make the whole angle equal to half of itself.",
            "Correct. Triangles HGJ and HIJ are congruent (HG ≅ HI by the Two Tangents Theorem, JG ≅ JI as radii, HJ shared), so HJ bisects ∠GHI. The full angle is therefore twice the half-angle: m∠GHI = 2θ. On the source figure, with a radius of 5 and a tangent of 8, θ = tan⁻¹(5/8) ≈ 32.005°, giving m∠GHI ≈ 64.01°.",
            "90° − θ is the measure of ∠GJH, the angle at the CENTRE in the same right triangle — the other acute angle, not the doubled one at H.",
            "GHIJ does have supplementary opposite angles, since the two right angles at G and I put it on a circle. But that gives ∠GHI = 180° − ∠GJI, and ∠GJI is the central angle, not θ."
          ],
          "hint": "Prove the two right triangles congruent first. What does that make the segment from H to the centre?"
        },
        {
          "id": "cta-011",
          "topic": "Sector Area",
          "difficulty": "hard",
          "stem": "A sector of a circle of radius 5 has a central angle of 57.995°. What is its area, to three decimal places?",
          "choices": [
            "6.983 units²",
            "78.540 units²",
            "5.061 units²",
            "2.531 units²"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A sector is the fraction of the circle its angle cuts out: A = (θ/360)·πr² = (57.995/360)·π(5²) = (57.995/360)(25π) ≈ 6.983 units².",
            "78.540 is the area of the WHOLE circle, π(5²) ≈ 78.54. The fraction θ/360 still has to be applied.",
            "5.061 is the ARC LENGTH for this angle, (57.995/360)(2π·5) ≈ 5.06. Arc length uses the circumference; area uses πr².",
            "2.531 is half the arc length, or the arc length computed with the radius rather than the diameter. Neither is an area — check the units the formula produces."
          ],
          "hint": "A sector is a fraction of a full circle. Work out which fraction, then apply it to the whole area."
        },
        {
          "id": "cta-012",
          "topic": "Circles on the Coordinate Plane",
          "difficulty": "medium",
          "stem": "The equation x² + y² − 6x − 8y + 9 = 16 describes a circle. After completing the square, what is its radius?",
          "choices": [
            "16 units",
            "5 units",
            "3 units",
            "4 units"
          ],
          "correctIndex": 3,
          "explanations": [
            "16 is r², not r. Standard form is (x − h)² + (y − k)² = r², so the number on the right is the SQUARE of the radius.",
            "5 units would follow from r² = 25. Completing the square here gives 16 on the right, not 25 — check the constants carried across.",
            "3 units is the x-coordinate of the centre, which is 3. The centre is (3, 4); neither coordinate is the radius.",
            "Correct. Group and complete: (x² − 6x + 9) + (y² − 8y + 16) = 16 − 9 + 9 + 16, giving (x − 3)² + (y − 4)² = 16. So r² = 16 and r = √16 = 4 units, centred at (3, 4)."
          ],
          "hint": "Complete the square in x and in y separately, remembering to add the same constants to the right-hand side. Then read r² off, and take the root."
        },
        {
          "id": "cta-013",
          "topic": "Chord-Chord Power",
          "difficulty": "medium",
          "stem": "Two chords intersect inside a circle. One is divided into segments of 3 and 8; the other into segments of x and 6. Find x.",
          "choices": [
            "4",
            "5",
            "11",
            "16"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The chord–chord form of the Power of a Point says the products of the two segments of each chord are equal: 3 · 8 = x · 6, so 24 = 6x and x = 4.",
            "5 comes from subtracting rather than using products: 11 − 6 = 5. The theorem is multiplicative, not additive — segment products are what stay equal.",
            "11 is the total length of the first chord, 3 + 8. Adding the segments of one chord tells you its length but says nothing directly about the other chord's pieces.",
            "16 comes from setting 3 · 8 = x · something and dividing by 3 rather than by 6. Match each factor to the correct chord before dividing."
          ],
          "hint": "Each chord contributes a product of its two pieces. Set the two products equal."
        },
        {
          "id": "cta-014",
          "topic": "Secant-Tangent Power",
          "difficulty": "hard",
          "stem": "From an external point, a tangent segment of length t and a secant are drawn to a circle. The secant's whole length is 16 and its external portion is 4. Find t.",
          "choices": [
            "4",
            "8",
            "12",
            "64"
          ],
          "correctIndex": 1,
          "explanations": [
            "4 is the external segment of the secant, given in the problem. It is one factor in the product, not the answer.",
            "Correct. The secant–tangent form of the Power of a Point gives t² = (whole) × (external) = 16 × 4 = 64, so t = √64 = 8.",
            "12 is the INTERNAL portion of the secant, 16 − 4. The theorem multiplies the whole secant by its external part, not the internal part.",
            "64 is t², not t. The final square root was skipped — a very common last-step slip on this theorem."
          ],
          "hint": "The tangent length appears squared in this relationship. Remember to undo that at the end."
        },
        {
          "id": "cta-015",
          "topic": "External Angle",
          "difficulty": "medium",
          "stem": "Two secants drawn from an external point intercept arcs of 120° and 30°. Find the measure of the angle at the external point.",
          "choices": [
            "45°",
            "75°",
            "90°",
            "150°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. An angle formed OUTSIDE a circle by two secants equals half the DIFFERENCE of the intercepted arcs: ½(120 − 30) = ½(90) = 45°.",
            "75° is half the SUM, ½(120 + 30). Half the sum is the rule for two chords meeting INSIDE the circle. Inside uses the sum; outside uses the difference.",
            "90° is the raw difference 120 − 30 without halving. Every one of these arc-angle rules carries a factor of ½.",
            "150° is the raw sum of the two arcs, with neither the subtraction nor the halving applied."
          ],
          "hint": "Position determines the operation: vertex inside the circle behaves one way, vertex outside the other. Both then halve."
        },
        {
          "id": "cta-016",
          "topic": "Angle in a Semicircle",
          "difficulty": "easy",
          "stem": "An angle is inscribed in a semicircle. What is its measure?",
          "choices": [
            "45°",
            "60°",
            "90°",
            "180°"
          ],
          "correctIndex": 2,
          "explanations": [
            "45° would require an intercepted arc of 90°, a quarter circle. A semicircle intercepts a 180° arc.",
            "60° would require a 120° intercepted arc. That is not what a semicircle provides.",
            "Correct. An inscribed angle is half its intercepted arc, and an angle inscribed in a semicircle intercepts the full 180° arc. Half of 180° is 90°, so the angle is always right — this is Thales' theorem, and it is why a diameter subtends a right angle from any point on the circle.",
            "180° is the ARC, not the inscribed angle. Confusing an arc with the inscribed angle that intercepts it drops the factor of ½."
          ],
          "hint": "Inscribed angle equals half the intercepted arc. Work out what arc a semicircle supplies."
        },
        {
          "id": "cta-017",
          "topic": "Two Chords Inside",
          "difficulty": "medium",
          "stem": "Two chords intersect inside a circle, intercepting arcs of 80° and 40°. Find the angle formed at their intersection.",
          "choices": [
            "20°",
            "40°",
            "60°",
            "120°"
          ],
          "correctIndex": 2,
          "explanations": [
            "20° is half the DIFFERENCE, ½(80 − 40). Half the difference is the rule for a vertex OUTSIDE the circle; this vertex is inside.",
            "40° is one of the given arcs restated. The angle depends on both arcs together, not on either one alone.",
            "Correct. An angle formed by two chords meeting inside a circle equals half the SUM of the two intercepted arcs: ½(80 + 40) = ½(120) = 60°.",
            "120° is the raw sum of the arcs with the halving step omitted."
          ],
          "hint": "The vertex sits inside the circle here. That determines whether you add or subtract the arcs before halving."
        },
        {
          "id": "cta-018",
          "topic": "Reasoning: Choose the Theorem",
          "difficulty": "hard",
          "stem": "Two chords of a circle are congruent. Which conclusion follows, and by what reasoning?",
          "choices": [
            "They are parallel, because congruent chords never meet.",
            "They are equidistant from the centre, because the perpendicular distance determines chord length in a given circle.",
            "They both pass through the centre.",
            "They intercept supplementary arcs."
          ],
          "correctIndex": 1,
          "explanations": [
            "Congruent chords can certainly intersect — think of two chords of equal length crossing near the centre.",
            "Correct. In a given circle, half-chord, distance-to-centre and radius form a right triangle. With the radius fixed, equal chords force equal distances from the centre — and the converse holds too.",
            "Only a diameter passes through the centre, and it is the longest chord. Congruent chords need not be diameters.",
            "Congruent chords intercept CONGRUENT arcs, not supplementary ones."
          ],
          "hint": "Draw the perpendicular from the centre to each chord. What triangle appears, and which of its sides are fixed?",
          "kind": "reasoning"
        },
        {
          "id": "cta-019",
          "topic": "Reasoning: What's Missing",
          "difficulty": "medium",
          "stem": "A student is given that an angle formed by two lines meeting a circle measures 35°, and is asked for the arcs. Why is that not yet answerable?",
          "choices": [
            "The radius is missing.",
            "Where the vertex sits — inside, on, or outside the circle — decides which relationship applies, and it has not been stated.",
            "The angle must be a whole number of degrees.",
            "You also need to know the circumference."
          ],
          "correctIndex": 1,
          "explanations": [
            "The radius has no bearing on angle-arc relationships. They are the same in a circle of any size.",
            "Correct. A vertex at the centre gives angle = arc; on the circle, half the arc; inside, half the SUM; outside, half the DIFFERENCE. Without knowing which, the same 35° gives four different answers.",
            "35° is already a whole number, and the relationships do not require it.",
            "Circumference is a length. Arc MEASURE in degrees is independent of the circle's size."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}