{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Points, Lines & Planes",
    "author": "Staples Education",
    "description": "Category 2 of the Geometry sequence: the undefined terms, betweenness and the Segment Addition Postulate, the distance formula, detour/surcharge bounds, the Triangle Inequality, and the counting facts that follow from two points determining a line. Converted from worked one-to-one tutoring sessions.",
    "created": "2026-08-06T04:14:55.936Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "points-lines-planes",
      "title": "Points, Lines & Planes",
      "blurb": "The undefined terms, betweenness and segment addition, distance and detour bounds, the Triangle Inequality, and what a pair of points actually determines.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "points-lines-planes"
      ],
      "questions": [
        {
          "id": "plp-001",
          "topic": "Distance and Detours",
          "difficulty": "medium",
          "stem": "Let P = (0, 0), Q = (4, 3), and M = (8, 0). A route P → Q → M has a surcharge δ equal to the extra distance travelled compared with going straight from P to M. Find δ(P → Q → M).",
          "choices": [
            "2",
            "10",
            "5",
            "0"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The surcharge is d(P,Q) + d(Q,M) − d(P,M). By the distance formula d(P,Q) = √(4² + 3²) = 5 and d(Q,M) = √(4² + 3²) = 5, while the direct route d(P,M) = 8. So δ = 5 + 5 − 8 = 2.",
            "10 is the total distance actually travelled, 5 + 5. The surcharge is the extra you paid over the direct route, so the direct distance still has to be subtracted.",
            "5 is the length of one leg, not the detour cost. Both legs contribute, and the straight-line distance comes off the total.",
            "0 would mean the detour cost nothing — which happens exactly when Q lies on segment PM. Here Q = (4,3) sits well above the x-axis that P and M lie on, so the route genuinely bends."
          ],
          "hint": "Add the two legs, then subtract what the direct trip would have cost. Both legs are 3-4-5 right triangles.",
          "diagram": "<svg viewBox=\"0 0 240 130\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"30\" y1=\"100\" x2=\"210\" y2=\"100\" stroke=\"#6b6b76\" stroke-width=\"1.6\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"100\" x2=\"120\" y2=\"30\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"30\" x2=\"210\" y2=\"100\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"30\" cy=\"100\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"120\" cy=\"30\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"210\" cy=\"100\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"18\" 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\">P (0,0)</text><text x=\"100\" y=\"22\" 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\">Q (4,3)</text><text x=\"186\" 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 (8,0)</text><text x=\"56\" y=\"60\" fill=\"#1f2024\" font-size=\"12\" 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=\"174\" y=\"60\" fill=\"#1f2024\" font-size=\"12\" 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=\"112\" y=\"116\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">8</text></svg>",
          "diagramCaption": "Solid: the route through Q. Dashed: the direct route. Not drawn to scale"
        },
        {
          "id": "plp-002",
          "topic": "Betweenness",
          "difficulty": "medium",
          "stem": "What does it mean geometrically if a waypoint Q is a “free waypoint”, so that δ(P → Q → M) = 0?",
          "choices": [
            "Q forms a right triangle with P and M.",
            "Q lies anywhere on the infinite line containing P and M.",
            "Q lies exactly on the line segment PM.",
            "Q is equidistant from P and M."
          ],
          "correctIndex": 2,
          "explanations": [
            "A right triangle still has three distinct vertices, so the two legs together are strictly longer than the hypotenuse and the surcharge is positive. A right angle at Q makes the detour efficient, not free.",
            "The infinite line is too generous. If Q sits on the line but beyond M, you overshoot and then double back — d(P,Q) + d(Q,M) exceeds d(P,M), so δ > 0. Only the portion between P and M is free.",
            "Correct. By the Straightness Law, the surcharge is zero if and only if the waypoint lies on the segment connecting the endpoints. This is the Segment Addition Postulate stated as a cost: PQ + QM = PM exactly when Q is between P and M.",
            "Equidistant points form the perpendicular bisector of PM, which is an entire line of points. Only one of them — the midpoint — is actually on the segment and therefore free; the rest cost extra."
          ],
          "hint": "Zero extra distance means the two legs add up to exactly the direct distance. Which points Q make PQ + QM = PM, and which make it more?",
          "diagram": "<svg viewBox=\"0 0 240 118\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"60\" x2=\"215\" y2=\"60\" stroke=\"#7c3aed\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><circle cx=\"25\" cy=\"60\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"215\" cy=\"60\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"120\" cy=\"60\" r=\"3.2\" fill=\"#1f2024\"/><text x=\"16\" y=\"48\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">P</text><text x=\"210\" y=\"48\" fill=\"#6b6b76\" font-size=\"12\" 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=\"112\" y=\"48\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">Q</text><text x=\"52\" y=\"82\" 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\">PQ</text><text x=\"154\" y=\"82\" 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\">QM</text><text x=\"86\" y=\"104\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">PQ + QM = PM</text></svg>",
          "diagramCaption": "A free waypoint sits on the segment itself. Not drawn to scale"
        },
        {
          "id": "plp-003",
          "topic": "Triangle Inequality",
          "difficulty": "easy",
          "stem": "Given three lengths — 5 cm, 7 cm, and 15 cm — why can these not form a valid triangle?",
          "choices": [
            "Because 15 is an odd number.",
            "Because they do not satisfy the Pythagorean theorem.",
            "Because they violate the Triangle Inequality: 5 + 7 < 15.",
            "Because 15 − 7 > 5."
          ],
          "correctIndex": 2,
          "explanations": [
            "Parity has nothing to do with it. Plenty of valid triangles have odd side lengths — 5, 7, 9 works perfectly well.",
            "The Pythagorean theorem is a test for right triangles specifically, not a requirement for triangles in general. Most triangles fail it and exist anyway.",
            "Correct. A triangle requires the sum of any two sides to be strictly greater than the third. Here 5 + 7 = 12, which is less than 15, so the two short sides cannot reach across the long one — they fall short by 3 cm no matter how you angle them.",
            "This inequality is true (8 > 5), and it is a real consequence of the same failure, but it is the difference form of the test rather than the reason. The direct statement is that the two shorter sides must together exceed the longest."
          ],
          "hint": "Try laying the two shorter sticks along the longest one. Do they reach?",
          "diagram": "<svg viewBox=\"0 0 240 106\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"80\" x2=\"205\" y2=\"80\" stroke=\"#6b6b76\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"55\" x2=\"85\" y2=\"55\" stroke=\"#7c3aed\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><line x1=\"85\" y1=\"55\" x2=\"169\" y2=\"55\" stroke=\"#7c3aed\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><line x1=\"169\" y1=\"55\" x2=\"205\" y2=\"55\" stroke=\"#c0392b\" stroke-width=\"1.6\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"94\" 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\">15 cm</text><text x=\"40\" y=\"44\" 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 cm</text><text x=\"112\" y=\"44\" 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\">7 cm</text><text x=\"160\" y=\"30\" fill=\"#c0392b\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">gap</text></svg>",
          "diagramCaption": "The two shorter sides laid end to end still fall short of the longest. Not drawn to scale"
        },
        {
          "id": "plp-004",
          "topic": "Distance and Detours",
          "difficulty": "hard",
          "stem": "The Surcharge Bound states that a detour can never cost more than twice the shorter of its two legs. If d(P, Q) = 4 and d(Q, M) = 9, what is the maximum possible surcharge δ(P → Q → M)?",
          "choices": [
            "13",
            "18",
            "8",
            "5"
          ],
          "correctIndex": 2,
          "explanations": [
            "13 is d(P,Q) + d(Q,M), the total distance travelled. The surcharge subtracts the direct distance from that total, and the direct distance is never zero here.",
            "18 is twice the LONGER leg. The bound is set by the shorter one — the worst case is doubling back along the short leg, which cannot cost more than that leg twice over.",
            "Correct. The bound is 2·min(d(P,Q), d(Q,M)) = 2 × 4 = 8. The worst case is the fully degenerate detour where Q lies on the far side of P: you walk 4 out, then 9 back past your start, and the excess over the direct 5 is exactly 8.",
            "5 is the difference 9 − 4, which is the SMALLEST the direct distance d(P,M) can be, not the largest the surcharge can be. It is the other end of the same triangle inequality."
          ],
          "hint": "The bound depends on only one of the two legs. Which one, and what multiple of it?"
        },
        {
          "id": "plp-005",
          "topic": "Points and Lines",
          "difficulty": "medium",
          "stem": "How many distinct straight lines can be drawn through 6 points in general position (no three of them collinear)?",
          "choices": [
            "15",
            "20",
            "30",
            "12"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Two points determine exactly one line, so the count is the number of ways to choose 2 points from 6: C(6,2) = 6·5/2 = 15. Counting by hand gives the same thing — the first point pairs with 5 others, the second with 4 new ones, and so on: 5 + 4 + 3 + 2 + 1 = 15.",
            "20 is C(6,3), the number of ways to choose three points — which counts triangles, not lines. A line is fixed by two points, and asking for a third over-specifies it.",
            "30 is 6 × 5, the number of ORDERED pairs. Every line gets counted twice that way, once from each endpoint, so the result has to be halved.",
            "12 is 6 × 2. There is no reading of the problem this corresponds to; the pairing structure is what governs the count."
          ],
          "hint": "How many points does it take to pin down a line? Then count how many such sets 6 points contain — remembering that order does not matter.",
          "diagram": "<svg viewBox=\"0 0 240 140\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"40\" y1=\"30\" x2=\"200\" y2=\"45\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"60\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"150\" y2=\"105\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"200\" y1=\"45\" x2=\"60\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"200\" y1=\"45\" x2=\"150\" y2=\"105\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"60\" y1=\"110\" x2=\"150\" y2=\"105\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"120\" y2=\"15\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"200\" y1=\"45\" x2=\"120\" y2=\"15\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"60\" y1=\"110\" x2=\"120\" y2=\"15\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"150\" y1=\"105\" x2=\"120\" y2=\"15\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"205\" y2=\"95\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"200\" y1=\"45\" x2=\"205\" y2=\"95\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"60\" y1=\"110\" x2=\"205\" y2=\"95\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"150\" y1=\"105\" x2=\"205\" y2=\"95\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"15\" x2=\"205\" y2=\"95\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><circle cx=\"40\" cy=\"30\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"200\" cy=\"45\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"60\" cy=\"110\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"150\" cy=\"105\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"120\" cy=\"15\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"205\" cy=\"95\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"96\" y=\"130\" 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\">6 points, every pair joined</text></svg>",
          "diagramCaption": "Every pair of points determines one line. Not drawn to scale"
        },
        {
          "id": "plp-006",
          "topic": "Points and Lines",
          "difficulty": "hard",
          "stem": "In a convex octagon (8 sides), how many interior crossing points do the diagonals create, assuming no three diagonals meet at the same point?",
          "choices": [
            "28",
            "70",
            "56",
            "120"
          ],
          "correctIndex": 1,
          "explanations": [
            "28 is C(8,2), the number of total possible connections between vertices — every side and diagonal counted once. That is a different quantity: it counts the segments themselves, not the points where they cross inside.",
            "Correct. Every interior crossing corresponds uniquely to a set of 4 vertices: choose any four, and the two diagonals of the quadrilateral they form cross exactly once. So the count is C(8,4) = (8·7·6·5)/(4·3·2·1) = 70.",
            "56 is 8·7, the number of ordered vertex pairs. Crossings are governed by sets of four vertices, not pairs.",
            "120 is C(10,3) and does not correspond to any structure in an octagon. Check which size of subset a single crossing determines."
          ],
          "hint": "Pick one crossing and ask how many vertices it takes to describe it. Then count subsets of that size."
        },
        {
          "id": "plp-007",
          "topic": "Points and Lines",
          "difficulty": "medium",
          "stem": "Using the diagonal formula d(n) = n(n − 3)/2, how many diagonals does a convex decagon (10 sides) have?",
          "choices": [
            "20",
            "35",
            "45",
            "50"
          ],
          "correctIndex": 1,
          "explanations": [
            "20 would come from n(n − 3)/2 with n = 8, or from halving 40. Substituting n = 10 gives 10 × 7, not 10 × 4.",
            "Correct. Substituting n = 10 gives d(10) = 10(10 − 3)/2 = 10 × 7 / 2 = 70/2 = 35. The formula itself is C(n,2) − n: all the segments joining vertices, minus the n that are sides.",
            "45 is C(10,2) = 45, every segment joining two vertices. That total still includes the 10 sides, which are not diagonals — subtracting them leaves 35.",
            "50 is 10 × 5, which would follow from using n − 5 in place of n − 3. Each vertex joins to n − 3 others by a diagonal: all the vertices except itself and its two neighbours."
          ],
          "hint": "Substitute n = 10 carefully, and divide by 2 only once — each diagonal is counted from both of its ends."
        },
        {
          "id": "plp-008",
          "topic": "Angle Basics",
          "difficulty": "easy",
          "stem": "Two adjacent angles form a straight line. If one angle measures 115°, what is the measure of the other?",
          "choices": [
            "65°",
            "75°",
            "25°",
            "115°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Two adjacent angles on a straight line form a linear pair, and a linear pair sums to 180°. So the other angle is 180° − 115° = 65°.",
            "75° would follow from subtracting 115 from 190. A straight line measures 180°, not 190°.",
            "25° is 90° − 65°, the complement of the correct answer. Complementary means summing to 90°, which describes a right angle rather than a straight line.",
            "115° repeats the given angle. Angles in a linear pair are equal only in the special case where both are 90°, and 115° is not 90°."
          ],
          "hint": "A straight line is a 180° angle, and the two adjacent angles divide it between them.",
          "diagram": "<svg viewBox=\"0 0 240 105\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"20\" y1=\"85\" x2=\"220\" y2=\"85\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"85\" x2=\"62\" y2=\"25\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"66\" y=\"76\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">115°</text><text x=\"140\" y=\"76\" fill=\"#6b6b76\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><circle cx=\"120\" cy=\"85\" r=\"3.2\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "The two angles form a linear pair. Not drawn to scale"
        },
        {
          "id": "plp-009",
          "topic": "Angle Basics",
          "difficulty": "easy",
          "stem": "What relationship always holds true for vertical angles?",
          "choices": [
            "They are congruent (equal in measure).",
            "They are supplementary.",
            "They are complementary.",
            "They share a common side."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Vertical angles sit opposite each other at an intersection of two lines, and they must be equal in measure. Each one is the supplement of the same adjacent angle, so the two are forced to match.",
            "Supplementary describes the ADJACENT pairs at that intersection — each vertical angle and the angle beside it. The opposite pair is equal, and equal pairs sum to 180° only in the perpendicular case.",
            "Complementary means summing to 90°. Nothing at a general intersection of two lines guarantees that.",
            "Vertical angles are opposite one another and share only the vertex point, not a side. The pairs that share a side are the adjacent, supplementary ones."
          ],
          "hint": "Both members of the pair are supplementary to the same neighbouring angle. What does that force?",
          "diagram": "<svg viewBox=\"0 0 240 122\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"25\" x2=\"215\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"105\" x2=\"215\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"108\" y=\"32\" 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\">a</text><text x=\"108\" y=\"108\" 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\">a</text><text x=\"40\" y=\"70\" fill=\"#6b6b76\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">b</text><text x=\"196\" y=\"70\" fill=\"#6b6b76\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">b</text><circle cx=\"120\" cy=\"65\" r=\"3.2\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "Opposite angles at an intersection are congruent. Not drawn to scale"
        },
        {
          "id": "plp-010",
          "topic": "Distance and Detours",
          "difficulty": "medium",
          "stem": "Point B lies between A and C on a line. If AB = 3x − 4, BC = x + 6, and AC = 22, what is the value of x?",
          "choices": [
            "5",
            "6",
            "8",
            "20"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Because B is between A and C, the Segment Addition Postulate gives AB + BC = AC. So (3x − 4) + (x + 6) = 22, which simplifies to 4x + 2 = 22, giving 4x = 20 and x = 5.",
            "6 comes from dropping the +2: solving 4x = 24 rather than 4x + 2 = 22. The constants −4 and +6 combine to +2, which has to move across before dividing.",
            "8 would follow from setting the two expressions equal to each other (3x − 4 = x + 6 gives x = 5, not 8) — and in any case B being between A and C does not make the two pieces equal. That would require B to be the midpoint.",
            "20 is 4x, a value that appears mid-solution but is not x itself. One more division is needed."
          ],
          "hint": "Betweenness is what licenses adding the two pieces. Write that equation before touching the algebra.",
          "diagram": "<svg viewBox=\"0 0 240 118\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"60\" x2=\"215\" y2=\"60\" stroke=\"#7c3aed\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><circle cx=\"25\" cy=\"60\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"130\" cy=\"60\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"215\" cy=\"60\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"18\" y=\"48\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"124\" y=\"48\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"210\" y=\"48\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"48\" y=\"82\" 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\">3x − 4</text><text x=\"146\" y=\"82\" 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\">x + 6</text><text x=\"104\" y=\"104\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">AC = 22</text></svg>",
          "diagramCaption": "B is between A and C. Not drawn to scale"
        },
        {
          "id": "plp-011",
          "topic": "Undefined Terms",
          "difficulty": "easy",
          "stem": "Point, line, and plane are called the “undefined terms” of geometry. Why are they left undefined?",
          "choices": [
            "Because mathematicians have not yet agreed on definitions for them.",
            "Because any definition would have to use other terms, which would need defining in turn — so the chain has to start somewhere.",
            "Because they cannot be drawn accurately.",
            "Because they only exist in the coordinate plane."
          ],
          "correctIndex": 1,
          "explanations": [
            "There is no outstanding disagreement here. Leaving them undefined is a deliberate structural choice made at the foundation of the subject, not an unfinished argument.",
            "Correct. Every definition is written in terms of something else. If nothing were taken as primitive, the definitions would either run forever or loop back on themselves. Point, line and plane are the agreed starting points: they are described by how they behave in the postulates rather than defined from anything prior.",
            "Drawing accuracy is a practical matter and has nothing to do with definability. A circle cannot be drawn perfectly either, and it is defined precisely.",
            "These terms predate coordinates by roughly two thousand years and apply just as well in synthetic geometry, where no coordinate system exists at all."
          ],
          "hint": "Try actually defining “point” without using any geometric word that would itself need defining."
        },
        {
          "id": "plp-012",
          "topic": "Undefined Terms",
          "difficulty": "medium",
          "stem": "Three points are chosen at random in space. Which statement is guaranteed to be true?",
          "choices": [
            "They always determine exactly one plane.",
            "They always lie on exactly one line.",
            "There is at least one plane containing all three.",
            "They always form a triangle."
          ],
          "correctIndex": 2,
          "explanations": [
            "“Exactly one” fails when the three points happen to be collinear: infinitely many planes contain a single line, so no one plane is picked out. The postulate requires the three points to be non-collinear before it promises uniqueness.",
            "Three points lie on one line only if they happen to be collinear, which random points generally are not. Two points always determine a line; three do not.",
            "Correct. Some plane always contains all three — if they are non-collinear it is unique, and if they are collinear there are infinitely many to choose from. Existence holds in both cases; only uniqueness needs the non-collinear condition.",
            "Collinear points form a degenerate figure with no area, not a triangle. A triangle needs three non-collinear vertices."
          ],
          "hint": "Separate two different claims — that such a plane exists, and that only one does. The collinear case breaks exactly one of them."
        },
        {
          "id": "plp-013",
          "topic": "Determining a Plane",
          "difficulty": "easy",
          "stem": "How many non-collinear points are required to determine exactly one plane?",
          "choices": [
            "Two",
            "Three",
            "Four",
            "Any number, as long as they are distinct"
          ],
          "correctIndex": 1,
          "explanations": [
            "Two points determine exactly one LINE, not a plane. Infinitely many planes contain any given line — picture a book's pages all sharing the spine.",
            "Correct. Three non-collinear points determine exactly one plane. The non-collinear condition is doing real work: three points on a single line still leave infinitely many planes containing them, which is why a three-legged stool never wobbles but its legs must not be in a row.",
            "Four points generally determine NO single plane — a fourth point usually sits off the plane fixed by the first three. Four points lying in one plane is a special condition called coplanarity, not a general fact.",
            "Distinctness is not enough. Three distinct but collinear points fail to pin down a plane, and more than three distinct points usually cannot lie in one plane at all."
          ],
          "hint": "Ask what two points determine first, then what has to be added — and what condition those points must satisfy."
        },
        {
          "id": "plp-014",
          "topic": "Intersecting Planes",
          "difficulty": "medium",
          "stem": "Two distinct planes intersect. What is their intersection?",
          "choices": [
            "A single point",
            "A line",
            "A plane",
            "They cannot intersect"
          ],
          "correctIndex": 1,
          "explanations": [
            "A single point would require the planes to touch and immediately separate, which flat unbounded surfaces cannot do. Once two planes share one point they necessarily share a whole line through it.",
            "Correct. If two distinct planes intersect at all, their intersection is exactly one line — the postulate behind the crease where two walls meet. Sharing a point forces sharing the entire line through it.",
            "Sharing a whole plane would make the two planes identical, contradicting the premise that they are distinct. Coincident planes are one plane, not two.",
            "Distinct planes certainly can intersect. The only planes that fail to intersect are parallel ones, which is a special case, not the general rule."
          ],
          "hint": "Think about where two walls of a room meet, and what shape that meeting is."
        },
        {
          "id": "plp-015",
          "topic": "Segment Addition",
          "difficulty": "easy",
          "stem": "Point B lies between A and C on a line. If AB = 3x − 1, BC = x + 5, and AC = 28, find x.",
          "choices": [
            "4",
            "6",
            "7",
            "8"
          ],
          "correctIndex": 1,
          "explanations": [
            "x = 4 gives AB = 11 and BC = 9, totalling 20 rather than 28. This comes from solving 4x + 4 = 20 — using the wrong total.",
            "Correct. By the Segment Addition Postulate, AB + BC = AC. So (3x − 1) + (x + 5) = 28, giving 4x + 4 = 28, then 4x = 24 and x = 6. Check: AB = 17, BC = 11, and 17 + 11 = 28.",
            "x = 7 gives AB = 20 and BC = 12, summing to 32 — an overshoot. This usually comes from dropping the constant +4 and solving 4x = 28.",
            "x = 8 gives AB = 23 and BC = 13, summing to 36. This is far past the given total and suggests the two expressions were combined incorrectly before solving."
          ],
          "hint": "Betweenness turns into addition: the two shorter pieces must sum to the whole."
        },
        {
          "id": "plp-016",
          "topic": "Midpoint",
          "difficulty": "medium",
          "stem": "A segment has midpoint (3, 4) and one endpoint at (1, 2). Find the other endpoint.",
          "choices": [
            "(2, 3)",
            "(5, 6)",
            "(4, 6)",
            "(−1, 0)"
          ],
          "correctIndex": 1,
          "explanations": [
            "(2, 3) is the midpoint of (1,2) and (3,4) — it treats the given midpoint as though it were the second endpoint. The midpoint is the destination of the averaging, not an input to it.",
            "Correct. The midpoint is the average of the endpoints, so each coordinate of the endpoint satisfies x = 2(3) − 1 = 5 and y = 2(4) − 2 = 6. Check: the midpoint of (1,2) and (5,6) is ((1+5)/2, (2+6)/2) = (3,4).",
            "(4, 6) has the right y but adds the run instead of doubling: from (1,2) the step to (3,4) is (+2,+2), so the far endpoint is (3+2, 4+2) = (5,6), not (4,6).",
            "(−1, 0) steps backwards from the given endpoint rather than forwards through the midpoint. It would put both given points on the same side of the answer."
          ],
          "hint": "The midpoint sits halfway. Whatever step takes you from the endpoint to the midpoint must be repeated once more."
        },
        {
          "id": "plp-017",
          "topic": "Collinearity",
          "difficulty": "hard",
          "stem": "Points A(1, 2), B(4, 8), and C(6, 12) are given. Which statement is true?",
          "choices": [
            "All three are collinear.",
            "Only A and B lie on a line through the origin.",
            "A, B and C form a triangle with positive area.",
            "B is the midpoint of AC."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Collinearity is tested by comparing slopes: AB = (8−2)/(4−1) = 2, BC = (12−8)/(6−4) = 2, and AC = (12−2)/(6−1) = 2. All three agree, and every point satisfies y = 2x, so the three lie on one line.",
            "A line through the origin has the form y = mx. Both A and B satisfy y = 2x — but so does C, since 2(6) = 12. The word 'only' is what makes this false; all three points are on that line.",
            "A triangle needs three NON-collinear vertices. Since all three slopes come out to 2, these points lie on a single line and enclose zero area — a degenerate triangle, not a real one.",
            "For B to be the midpoint of AC it would have to sit at ((1+6)/2, (2+12)/2) = (3.5, 7). B is at (4, 8), so it lies on the segment but not at its middle."
          ],
          "hint": "Compute the slope between each pair of points. Equal slopes throughout mean the points share a single line."
        },
        {
          "id": "plp-018",
          "topic": "Reasoning: Definitions",
          "difficulty": "medium",
          "stem": "A student says three points always determine exactly one plane. When is that claim false?",
          "choices": [
            "Never — three points always determine exactly one plane.",
            "When the three points are collinear, in which case infinitely many planes contain them.",
            "When the points are very far apart.",
            "When two of the points coincide, in which case no plane exists."
          ],
          "correctIndex": 1,
          "explanations": [
            "The postulate requires the three points to be NON-collinear, which the student's statement leaves out.",
            "Correct. Three collinear points lie on a single line, and infinitely many planes can be rotated about that line. The postulate is that three NON-collinear points determine exactly one plane.",
            "Distance is irrelevant. A plane is unbounded, so points can be any distance apart.",
            "Two coincident points still lie in planes — infinitely many, since it reduces to two distinct points at most."
          ],
          "hint": "State the postulate exactly. Which word is the student leaving out?",
          "kind": "reasoning"
        },
        {
          "id": "plp-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "Given that M is between A and B with AM = 7 and MB = 5, a student writes AB = 7 − 5 = 2. What is the error?",
          "choices": [
            "Betweenness means the parts ADD to the whole: AB = 7 + 5 = 12.",
            "The student should have used the midpoint formula.",
            "AB cannot be found without coordinates.",
            "Nothing is wrong; AB = 2."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Segment Addition Postulate says that if M is between A and B, then AM + MB = AB. So AB = 12.",
            "The midpoint formula finds a location from coordinates. M is not stated to be a midpoint here, and no coordinates are given.",
            "Coordinates are not needed. The two part-lengths and the betweenness relationship are enough.",
            "AB = 2 would be shorter than AM alone, which is impossible when M lies between A and B."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}