{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Coordinate Geometry",
    "author": "Staples Education",
    "description": "Distance, midpoint, slope, line equations, parallel and perpendicular conditions, and segment partitioning. Transcribed from the worked Coordinate Geometry section of the tutoring whiteboard archive.",
    "created": "2026-08-06T04:14:56.148Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "coordinate-geometry",
      "title": "Coordinate Geometry",
      "blurb": "Distance, midpoint, slope, writing line equations, parallel and perpendicular conditions, and partitioning a segment in a given ratio.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "coordinate-lines"
      ],
      "questions": [
        {
          "id": "cog-001",
          "topic": "Distance Formula",
          "difficulty": "easy",
          "stem": "Find the distance between the points (1, 2) and (4, 6).",
          "choices": [
            "5",
            "7",
            "√7",
            "25"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The distance formula gives d = √((4 − 1)² + (6 − 2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5. The 3-4-5 right triangle is doing the work here.",
            "7 adds the two differences, 3 + 4. That is the distance you would travel going across then up — the straight-line distance is the hypotenuse of those two legs, which is shorter.",
            "√7 comes from √(4 − 1 + 6 − 2) — adding the differences before squaring. Each difference has to be squared first.",
            "25 is d², one step short. The square root still has to be taken."
          ],
          "hint": "The differences in x and y are the legs of a right triangle. The distance is its hypotenuse.",
          "diagram": "<svg viewBox=\"0 0 170 140\" width=\"100%\" style=\"max-width:170px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"30\" y1=\"-6\" x2=\"30\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"48\" y1=\"-6\" x2=\"48\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"66\" y1=\"-6\" x2=\"66\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"84\" y1=\"-6\" x2=\"84\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"102\" y1=\"-6\" x2=\"102\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"-6\" x2=\"120\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"138\" y1=\"-6\" x2=\"138\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"120\" x2=\"138\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"102\" x2=\"138\" y2=\"102\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"84\" x2=\"138\" y2=\"84\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"66\" x2=\"138\" y2=\"66\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"48\" x2=\"138\" y2=\"48\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"30\" x2=\"138\" y2=\"30\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"12\" x2=\"138\" y2=\"12\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"-6\" x2=\"138\" y2=\"-6\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"120\" x2=\"138\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"-6\" x2=\"30\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"48\" y1=\"84\" x2=\"102\" y2=\"12\" stroke=\"#7c3aed\" stroke-width=\"2.2\" stroke-linecap=\"round\"/><line x1=\"48\" y1=\"84\" x2=\"102\" y2=\"84\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"102\" y1=\"84\" x2=\"102\" y2=\"12\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><circle cx=\"48\" cy=\"84\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"102\" cy=\"12\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"20\" y=\"96\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">(1,2)</text><text x=\"96\" y=\"20\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">(4,6)</text><text x=\"60\" 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\">3</text><text x=\"106\" y=\"62\" 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></svg>",
          "diagramCaption": "The horizontal and vertical gaps are the legs. Not drawn to scale"
        },
        {
          "id": "cog-002",
          "topic": "Midpoint",
          "difficulty": "easy",
          "stem": "Find the midpoint of the segment joining (−3, 5) and (7, −1).",
          "choices": [
            "(2, 2)",
            "(5, 3)",
            "(4, 4)",
            "(10, −6)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The midpoint is the average of the coordinates: ((−3 + 7)/2, (5 + (−1))/2) = (4/2, 4/2) = (2, 2).",
            "(5, 3) averages the absolute values, or halves the differences rather than the sums. The midpoint formula adds the coordinates before halving.",
            "(4, 4) is the sum of each pair without halving. Those sums are 4 and 4, which then have to be divided by 2.",
            "(10, −6) is the difference of the coordinates — the run and the rise. That is the vector from one point to the other, not the point between them."
          ],
          "hint": "A midpoint is an average. Average the x's, then average the y's."
        },
        {
          "id": "cog-003",
          "topic": "Slope",
          "difficulty": "easy",
          "stem": "Find the slope of the line through (2, 3) and (6, 11).",
          "choices": [
            "2",
            "1/2",
            "4",
            "8"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Slope is Δy/Δx = (11 − 3)/(6 − 2) = 8/4 = 2.",
            "1/2 inverts the ratio, computing Δx/Δy = 4/8. Slope is rise over run, not run over rise.",
            "4 is Δx alone, the horizontal change. Slope compares it to the vertical change.",
            "8 is Δy alone, the vertical change. It still has to be divided by the run."
          ],
          "hint": "Rise over run. Subtract in the same order on top and bottom."
        },
        {
          "id": "cog-004",
          "topic": "Equations of Lines",
          "difficulty": "medium",
          "stem": "Write the equation of the line through (0, −4) with slope 3.",
          "choices": [
            "y = 3x − 4",
            "y = −4x + 3",
            "y = 3x + 4",
            "y = 4x − 3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Point-slope gives y − y₁ = m(x − x₁), so y − (−4) = 3(x − 0), i.e. y = 3x − 4. Because the point is on the y-axis, −4 is the y-intercept and slope-intercept form can be written down directly.",
            "y = −4x + 3 swaps the slope and the intercept. The 3 is the slope and the −4 is the intercept, not the other way round.",
            "y = 3x + 4 has the right slope but the wrong sign on the intercept. The point is (0, −4), which sits below the origin.",
            "y = 4x − 3 also swaps the two numbers, keeping the signs. Check which one multiplies x."
          ],
          "hint": "The given point has x = 0, which means its y-coordinate IS the y-intercept."
        },
        {
          "id": "cog-005",
          "topic": "Parallel and Perpendicular",
          "difficulty": "medium",
          "stem": "Line A has slope 2/3. What is the slope of any line perpendicular to it?",
          "choices": [
            "2/3",
            "−3/2",
            "3/2",
            "−2/3"
          ],
          "correctIndex": 1,
          "explanations": [
            "2/3 repeats the given slope. Equal slopes mean PARALLEL lines, not perpendicular ones.",
            "Correct. Perpendicular slopes are negative reciprocals: m₁m₂ = −1, so m₂ = −1/(2/3) = −3/2. Flip the fraction and change the sign — both steps are needed.",
            "3/2 flips the fraction but keeps the sign positive. That product is (2/3)(3/2) = 1, not −1.",
            "−2/3 changes the sign but does not flip. That product is −4/9, not −1."
          ],
          "hint": "Two steps, not one: invert the fraction AND negate it. Check by multiplying — you should get −1."
        },
        {
          "id": "cog-006",
          "topic": "Parallel and Perpendicular",
          "difficulty": "medium",
          "stem": "Are the line through (0, 0) and (2, 4) and the line through (1, 1) and (3, 5) parallel?",
          "choices": [
            "Yes — both have slope 2.",
            "No — their slopes are 2 and 4.",
            "No — they intersect at the origin.",
            "Yes — they both pass through (1, 1)."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The first slope is (4 − 0)/(2 − 0) = 4/2 = 2. The second is (5 − 1)/(3 − 1) = 4/2 = 2. Equal slopes and different intercepts, so the lines are parallel.",
            "The second slope is 4/2 = 2, not 4. The rise of 4 still has to be divided by the run of 2.",
            "The first line passes through the origin; the second does not (at x = 0 it gives y = −1). And even if they did meet, that would disprove parallelism rather than explain it.",
            "The second line passes through (1, 1); the first does not — at x = 1 it gives y = 2. Sharing a point would in fact rule out parallel lines."
          ],
          "hint": "Compute both slopes. Parallel is decided by slope alone, nothing else."
        },
        {
          "id": "cog-007",
          "topic": "Distance Formula",
          "difficulty": "medium",
          "stem": "Find the length of the segment from (−2, −3) to (4, 5).",
          "choices": [
            "10",
            "14",
            "√14",
            "100"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Δx = 4 − (−2) = 6 and Δy = 5 − (−3) = 8, so d = √(36 + 64) = √100 = 10. Another 3-4-5 triangle, scaled by 2.",
            "14 adds the legs, 6 + 8. The straight-line distance is the hypotenuse, which is always shorter than going around two legs.",
            "√14 adds the differences before squaring. Square each difference first, then add.",
            "100 is d². Take the square root."
          ],
          "hint": "Watch the signs when subtracting negatives — both differences here come out positive."
        },
        {
          "id": "cog-008",
          "topic": "Midpoint",
          "difficulty": "hard",
          "stem": "A segment has midpoint (3, 4) and one endpoint (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) — the average of the endpoint and the midpoint. That is halfway to the halfway point.",
            "Correct. Set the midpoint formula equal to what is known: (1 + x)/2 = 3 gives x = 5, and (2 + y)/2 = 4 gives y = 6. So the other endpoint is (5, 6). Equivalently, step the same distance again: from (1,2) to (3,4) is +2 and +2, so continue to (5,6).",
            "(4, 6) has the right y but the wrong x. From 1, doubling the midpoint's 3 gives 6, minus 1 gives 5.",
            "(−1, 0) steps backwards from the known endpoint instead of forwards through the midpoint."
          ],
          "hint": "The midpoint is the average of the two ends. Write that equation with the unknown in it and solve, or just step the same distance past the midpoint."
        },
        {
          "id": "cog-009",
          "topic": "Equations of Lines",
          "difficulty": "medium",
          "stem": "Write, in slope-intercept form, the equation of the line through (2, 1) and (4, 7).",
          "choices": [
            "y = 3x − 5",
            "y = 3x + 5",
            "y = 2x − 3",
            "y = x/3 − 5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The slope is (7 − 1)/(4 − 2) = 6/2 = 3. Using point-slope at (2, 1): y − 1 = 3(x − 2), so y = 3x − 6 + 1 = 3x − 5.",
            "y = 3x + 5 has the right slope but the wrong intercept sign. Substituting x = 2 gives 11, not 1.",
            "y = 2x − 3 uses a slope of 2. The rise is 6 over a run of 2, which is 3.",
            "y = x/3 − 5 inverts the slope. Rise over run is 6/2, not 2/6."
          ],
          "hint": "Slope first, then substitute either point to pin down the intercept — and check your answer against the other point."
        },
        {
          "id": "cog-010",
          "topic": "Parallel and Perpendicular",
          "difficulty": "hard",
          "stem": "Find the equation of the line through (1, 5) that is perpendicular to y = ½x + 3.",
          "choices": [
            "y = ½x + 4.5",
            "y = −2x + 7",
            "y = 2x + 3",
            "y = −½x + 5.5"
          ],
          "correctIndex": 1,
          "explanations": [
            "This is parallel to the given line, not perpendicular — it reuses the slope ½ instead of taking the negative reciprocal.",
            "Correct. The given slope is ½, so the perpendicular slope is −2. Through (1, 5): 5 = −2(1) + b gives b = 7, so y = −2x + 7.",
            "y = 2x + 3 flips the slope but keeps it positive. The product (½)(2) = 1, not −1.",
            "y = −½x + 5.5 negates the slope without inverting it. The product (½)(−½) = −¼, not −1."
          ],
          "hint": "Negative reciprocal for the slope, then use the point to find b."
        },
        {
          "id": "cog-011",
          "topic": "Partitioning Segments",
          "difficulty": "hard",
          "stem": "Point P partitions the segment from A(0, 0) to B(10, 5) in the ratio 2:3 (measured from A). Find P.",
          "choices": [
            "(4, 2)",
            "(6, 3)",
            "(5, 2.5)",
            "(2, 3)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A 2:3 split puts P two-fifths of the way from A to B. So P = (0 + (2/5)(10), 0 + (2/5)(5)) = (4, 2). The ratio's parts total 5, which is the denominator — not 2 or 3 on their own.",
            "(6, 3) is three-fifths of the way, which is the 3:2 split — measured from B rather than from A.",
            "(5, 2.5) is the midpoint, which would be a 1:1 ratio. A 2:3 split is not halfway.",
            "(2, 3) reads the ratio itself as coordinates. The numbers 2 and 3 describe how the segment is divided, not where the point lands."
          ],
          "hint": "A 2:3 ratio divides the segment into 5 equal parts, and P sits after 2 of them.",
          "diagram": "<svg viewBox=\"0 0 240 104\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"70\" x2=\"215\" y2=\"70\" stroke=\"#7c3aed\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><circle cx=\"25\" cy=\"70\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"101\" cy=\"70\" r=\"3.2\" fill=\"#c0392b\"/><circle cx=\"215\" cy=\"70\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"16\" y=\"58\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A(0,0)</text><text x=\"86\" y=\"58\" 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\">P</text><text x=\"186\" y=\"58\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B(10,5)</text><text x=\"50\" y=\"90\" 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\">2 parts</text><text x=\"136\" y=\"90\" 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\">3 parts</text></svg>",
          "diagramCaption": "P divides AB in the ratio 2 : 3 from A. Not drawn to scale"
        },
        {
          "id": "cog-012",
          "topic": "Coordinate Proof",
          "difficulty": "medium",
          "stem": "Is the triangle with vertices (0, 0), (4, 0) and (0, 3) a right triangle?",
          "choices": [
            "Yes — the legs lie on the axes, which meet at a right angle.",
            "No — its sides are 3, 4 and 5, which is scalene.",
            "No — none of its angles is 90°.",
            "Only if the vertices are relabelled."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. One side runs along the x-axis and another along the y-axis, and the axes are perpendicular, so the angle at (0, 0) is exactly 90°. The side lengths confirm it: 3, 4 and 5, with 3² + 4² = 5².",
            "Scalene and right are not alternatives — they describe different things. A triangle is classified by its sides AND its angles, and a 3-4-5 triangle is both scalene and right.",
            "The angle at the origin is 90°, formed by the two axes. Checking the slopes gives the same answer: one side has slope 0, the other is vertical.",
            "Relabelling vertices does not change a figure's shape. The right angle is at (0, 0) however the points are named."
          ],
          "hint": "Look at where the sides lie before computing anything. Two of them are along the axes.",
          "diagram": "<svg viewBox=\"0 0 160 140\" width=\"100%\" style=\"max-width:160px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"40\" y1=\"30\" x2=\"40\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"60\" y1=\"30\" x2=\"60\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"80\" y1=\"30\" x2=\"80\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"100\" y1=\"30\" x2=\"100\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"30\" x2=\"120\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"140\" y1=\"30\" x2=\"140\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"110\" x2=\"140\" y2=\"110\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"90\" x2=\"140\" y2=\"90\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"70\" x2=\"140\" y2=\"70\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"50\" x2=\"140\" y2=\"50\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"140\" y2=\"30\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"110\" x2=\"140\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"30\" x2=\"40\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><polygon points=\"40,110 140,110 40,50\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><path d=\"M 40,98 L 52,98 L 52,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><circle cx=\"40\" cy=\"110\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"120\" cy=\"110\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"40\" cy=\"50\" r=\"3.2\" fill=\"#7c3aed\"/><text x=\"84\" 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\">4</text><text x=\"24\" y=\"84\" 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\">3</text><text x=\"96\" y=\"74\" 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></svg>",
          "diagramCaption": "Legs on the axes. Not drawn to scale"
        },
        {
          "id": "cog-013",
          "topic": "Distance Formula",
          "difficulty": "easy",
          "stem": "Find the distance between the points (−2, −3) and (4, 5).",
          "choices": [
            "10",
            "14",
            "√14",
            "√28"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The horizontal change is 4 − (−2) = 6 and the vertical change is 5 − (−3) = 8. Then d = √(6² + 8²) = √(36 + 64) = √100 = 10 — the 6-8-10 triple, a scaled 3-4-5.",
            "14 is 6 + 8, adding the two changes instead of combining them through the Pythagorean Theorem. That would be the walking distance along the grid, not the straight-line distance.",
            "√14 comes from adding the differences before squaring rather than squaring each first. The formula squares each difference individually, then adds.",
            "√28 comes from mishandling the negative coordinates — probably 4 − 2 = 2 and 5 − 3 = 2 instead of subtracting the negatives properly. Subtracting a negative adds."
          ],
          "hint": "Watch the signs when subtracting negative coordinates, then look for a familiar Pythagorean triple."
        },
        {
          "id": "cog-014",
          "topic": "Perpendicular Lines",
          "difficulty": "medium",
          "stem": "Find the equation of the line through (1, 5) that is perpendicular to y = ½x + 3.",
          "choices": [
            "y = ½x + 4.5",
            "y = −½x + 5.5",
            "y = −2x + 7",
            "y = 2x + 3"
          ],
          "correctIndex": 2,
          "explanations": [
            "This keeps the original slope of ½, which produces a PARALLEL line, not a perpendicular one. Parallel lines have equal slopes; perpendicular lines do not.",
            "This negates the slope but does not take the reciprocal. The perpendicular slope must satisfy m₁m₂ = −1, and (½)(−½) = −¼, not −1.",
            "Correct. The perpendicular slope is the negative reciprocal of ½, which is −2. Using point-slope through (1, 5): y − 5 = −2(x − 1), so y = −2x + 2 + 5 = −2x + 7. Check: (½)(−2) = −1.",
            "This takes the reciprocal but not the negative. A slope of 2 gives (½)(2) = 1, not −1, so the lines are not perpendicular."
          ],
          "hint": "Perpendicular slopes multiply to −1. That requires two changes to the original slope, not one."
        },
        {
          "id": "cog-015",
          "topic": "Partitioning a Segment",
          "difficulty": "hard",
          "stem": "Point P partitions the segment from A(0, 0) to B(10, 5) in the ratio 2:3, measured from A. Find P.",
          "choices": [
            "(4, 2)",
            "(6, 3)",
            "(5, 2.5)",
            "(2, 3)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A ratio of 2:3 splits the segment into 5 equal parts, and P sits 2 of them from A — that is ⅖ of the way. So P = (0 + ⅖·10, 0 + ⅖·5) = (4, 2).",
            "(6, 3) is ⅗ of the way, which measures the 3-part from A instead of the 2-part. The ratio is read starting from the first named endpoint.",
            "(5, 2.5) is the MIDPOINT, which corresponds to a 1:1 ratio. A 2:3 split is deliberately off-center.",
            "(2, 3) uses the ratio numbers directly as coordinates. The ratio determines a fraction of the total displacement, not the coordinates themselves."
          ],
          "hint": "A ratio of a:b divides the segment into a + b equal parts. Work out what fraction of the way P sits."
        },
        {
          "id": "cog-016",
          "topic": "Slope Through Two Points",
          "difficulty": "easy",
          "stem": "Write, in slope-intercept form, the equation of the line through (2, 1) and (4, 7).",
          "choices": [
            "y = 3x − 5",
            "y = 3x + 1",
            "y = ⅓x + ⅓",
            "y = 2x − 3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The slope is (7 − 1)/(4 − 2) = 6/2 = 3. Using point (2, 1): 1 = 3(2) + b, so b = 1 − 6 = −5, giving y = 3x − 5. Check with (4, 7): 3(4) − 5 = 7. ✓",
            "The slope is right but the intercept has the wrong sign. Substituting (2,1) gives b = 1 − 6 = −5, not +1.",
            "This inverts the slope, computing run over rise instead of rise over run. Slope is the change in y divided by the change in x.",
            "A slope of 2 comes from subtracting the coordinates in mismatched order, such as (4 − 2)/(7 − 1) inverted, or from using the wrong pair of values."
          ],
          "hint": "Find the slope first, then substitute either point back to solve for the intercept — and verify with the other point."
        },
        {
          "id": "cog-017",
          "topic": "Classifying by Slope",
          "difficulty": "medium",
          "stem": "Are the line through (0, 0) and (2, 4) and the line through (1, 1) and (3, 5) parallel, perpendicular, or neither?",
          "choices": [
            "Parallel",
            "Perpendicular",
            "Neither",
            "They are the same line"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The first line has slope (4 − 0)/(2 − 0) = 2 and the second has slope (5 − 1)/(3 − 1) = 2. Equal slopes with different intercepts means the lines are parallel and never meet.",
            "Perpendicular would require the slopes to multiply to −1. Here 2 × 2 = 4, so the lines are not perpendicular.",
            "'Neither' would apply if the slopes were unequal and their product were not −1. Both slopes are exactly 2, so the parallel relationship does hold.",
            "The same line would require identical slope AND identical intercept. The first passes through the origin (b = 0) while the second gives 1 = 2(1) + b, so b = −1. Different intercepts, so they are distinct."
          ],
          "hint": "Compute both slopes. If they match, check the intercepts too before concluding the lines are merely parallel."
        },
        {
          "id": "cog-018",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "A student says two segments are perpendicular because their slopes are 3 and −3. What is the actual test, and are they perpendicular?",
          "choices": [
            "Perpendicular slopes are opposite reciprocals, so the slope would need to be −1/3. They are not perpendicular.",
            "Perpendicular slopes are simply opposite in sign, so they are perpendicular.",
            "Perpendicular slopes must multiply to 1, so they are not perpendicular.",
            "Slope cannot determine perpendicularity."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Perpendicular slopes multiply to −1, which makes them opposite RECIPROCALS. The partner for 3 is −1/3, not −3. Here 3 × (−3) = −9.",
            "Opposite signs alone are not enough — the reciprocal is the part being dropped.",
            "The product must be −1, not 1. The conclusion happens to be right, but from a wrong rule.",
            "Slope determines it perfectly well for non-vertical lines; the test is just being misremembered."
          ],
          "hint": "Multiply the two slopes. What should the product be?",
          "kind": "reasoning"
        },
        {
          "id": "cog-019",
          "topic": "Reasoning: Choose the Method",
          "difficulty": "medium",
          "stem": "To prove a quadrilateral on the coordinate plane is a rhombus, which set of calculations is sufficient?",
          "choices": [
            "The slopes of all four sides, showing opposite sides parallel.",
            "The lengths of all four sides, showing all four congruent.",
            "The midpoints of the diagonals only.",
            "The area, compared against a known rhombus."
          ],
          "correctIndex": 1,
          "explanations": [
            "Parallel opposite sides prove a PARALLELOGRAM. A rhombus needs the sides to be equal in length as well.",
            "Correct. Four congruent sides is the definition of a rhombus, and the distance formula establishes it directly.",
            "Matching diagonal midpoints proves the diagonals bisect each other, which gives a parallelogram — not enough for a rhombus.",
            "Two very different quadrilaterals can share an area, so area proves nothing about shape."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}