{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Pythagoras & Right-Triangle Trigonometry",
    "author": "Staples Education",
    "description": "Pythagorean triples, special right triangles, the acute/right/obtuse classification test, the geometric-mean altitude, and applied SOH-CAH-TOA. Transcribed from the worked Pythagoras and SOHCAHTOA sections of the tutoring whiteboard archive.",
    "created": "2026-08-06T04:14:56.154Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "pythagorean-trig",
      "title": "Pythagoras & Right-Triangle Trigonometry",
      "blurb": "The Pythagorean theorem and its triples, the 30-60-90 and 45-45-90 families, the classification test, and SOH-CAH-TOA applied to real measurements.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "pythagorean-theorem",
        "right-triangle-trig",
        "triangles-review"
      ],
      "questions": [
        {
          "id": "pyt-001",
          "topic": "Pythagorean Theorem",
          "difficulty": "easy",
          "stem": "A right triangle has legs of length 9 and 12. Find the hypotenuse.",
          "choices": [
            "15",
            "21",
            "√21",
            "225"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15. This is the 3-4-5 triple scaled by 3.",
            "21 adds the legs. The hypotenuse is always shorter than the sum of the two legs — that is the Triangle Inequality.",
            "√21 adds the legs before squaring. Each leg has to be squared first.",
            "225 is c². The square root is the last step."
          ],
          "hint": "Square each leg, add, then take the root. Look for a familiar triple.",
          "diagram": "<svg viewBox=\"0 0 220 140\" width=\"100%\" style=\"max-width:220px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"30,110 190,110 30,30\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><path d=\"M 30,98 L 42,98 L 42,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"100\" y=\"128\" 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\">12</text><text x=\"14\" y=\"74\" 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\">9</text><text x=\"118\" y=\"66\" 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\">c</text></svg>",
          "diagramCaption": "Legs 9 and 12. Not drawn to scale"
        },
        {
          "id": "pyt-002",
          "topic": "Special Right Triangles",
          "difficulty": "medium",
          "stem": "The hypotenuse of a 45-45-90 triangle is 10. Find the length of each leg.",
          "choices": [
            "5",
            "10√2",
            "5√2",
            "10/√3"
          ],
          "correctIndex": 2,
          "explanations": [
            "5 halves the hypotenuse. That is the short-leg rule for a 30-60-90 triangle, not a 45-45-90 one.",
            "10√2 multiplies by √2 instead of dividing. The legs of a 45-45-90 triangle are shorter than its hypotenuse, so multiplying by √2 ≈ 1.41 goes the wrong way.",
            "Correct. In a 45-45-90 triangle the sides are in the ratio x : x : x√2, so leg = hypotenuse/√2 = 10/√2 = 5√2 ≈ 7.07 after rationalizing.",
            "10/√3 uses the 30-60-90 ratio. The √3 belongs to that family; the 45-45-90 family uses √2."
          ],
          "hint": "The ratio is x : x : x√2. You have the x√2 part and want x.",
          "diagram": "<svg viewBox=\"0 0 200 140\" width=\"100%\" style=\"max-width:200px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"30,110 170,110 30,30\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><path d=\"M 30,98 L 42,98 L 42,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"120\" y=\"62\" 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\">10</text><text x=\"88\" y=\"128\" 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\">x</text><text x=\"14\" y=\"74\" 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\">x</text><text x=\"126\" y=\"104\" 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\">45°</text><text x=\"36\" y=\"46\" 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\">45°</text></svg>",
          "diagramCaption": "A 45-45-90 triangle. Not drawn to scale"
        },
        {
          "id": "pyt-003",
          "topic": "Special Right Triangles",
          "difficulty": "medium",
          "stem": "In a 30-60-90 triangle the short leg is 6. Find the long leg and the hypotenuse.",
          "choices": [
            "long leg 6√3, hypotenuse 12",
            "long leg 12, hypotenuse 6√3",
            "long leg 6√2, hypotenuse 12",
            "long leg 3√3, hypotenuse 6"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The 30-60-90 ratio is x : x√3 : 2x with x the short leg. With x = 6 the long leg is 6√3 ≈ 10.39 and the hypotenuse is 12.",
            "This swaps the two. The hypotenuse must be the longest side, and 6√3 ≈ 10.39 is less than 12 — so 12 cannot be the long leg.",
            "6√2 uses the 45-45-90 constant. The 30-60-90 family uses √3.",
            "3√3 and 6 halve the short leg first. The ratio is built FROM the short leg, so x = 6 is used directly."
          ],
          "hint": "Write the ratio x : x√3 : 2x and identify which position the given 6 occupies."
        },
        {
          "id": "pyt-004",
          "topic": "Trigonometry",
          "difficulty": "medium",
          "stem": "A 12-ft ladder leans against a wall at a 65° angle of elevation. How high up the wall does it reach? (sin 65° ≈ 0.906)",
          "choices": [
            "10.87 ft",
            "13.24 ft",
            "5.07 ft",
            "11.44 ft"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The wall height is the side opposite the 65° angle and the ladder is the hypotenuse, so sin 65° = h/12, giving h = 12(0.906) ≈ 10.87 ft.",
            "13.24 divides by the sine instead of multiplying: 12/0.906. That would make the height longer than the ladder, which is impossible.",
            "5.07 ft uses cos 65° ≈ 0.423, giving the distance from the wall along the ground — the ADJACENT side, not the height.",
            "11.44 ft does not follow from either ratio. Check which side the 65° angle is opposite."
          ],
          "hint": "SOH-CAH-TOA. The ladder is the hypotenuse; the height is opposite the angle at the ground.",
          "diagram": "<svg viewBox=\"0 0 230 138\" width=\"100%\" style=\"max-width:230px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"30\" y1=\"120\" x2=\"200\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.8\" stroke-linecap=\"round\"/><line x1=\"200\" y1=\"120\" x2=\"200\" y2=\"30\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"120\" x2=\"200\" y2=\"30\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 200,108 L 188,108 L 188,120\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"48\" 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\">65°</text><text x=\"96\" y=\"66\" 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\">12 ft</text><text x=\"206\" y=\"78\" fill=\"#7c3aed\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">h</text></svg>",
          "diagramCaption": "The ladder makes 65° with the ground. Not drawn to scale"
        },
        {
          "id": "pyt-005",
          "topic": "Right Triangle Altitudes",
          "difficulty": "hard",
          "stem": "An altitude drawn to the hypotenuse of a right triangle splits it into segments of 4 and 9. Find the altitude.",
          "choices": [
            "6",
            "6.5",
            "13",
            "36"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The altitude to the hypotenuse is the geometric mean of the two segments: h = √(4 × 9) = √36 = 6.",
            "6.5 is the arithmetic mean, (4 + 9)/2. This relationship is a geometric mean — multiply and take the root.",
            "13 is the whole hypotenuse, 4 + 9.",
            "36 is h². One square root short."
          ],
          "hint": "The two small triangles are similar to each other, which gives h/4 = 9/h."
        },
        {
          "id": "pyt-006",
          "topic": "Pythagorean Theorem",
          "difficulty": "easy",
          "stem": "A right triangle has hypotenuse 13 and one leg 5. Find the other leg.",
          "choices": [
            "12",
            "18",
            "√194",
            "8"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Rearranged, b² = c² − a² = 13² − 5² = 169 − 25 = 144, so b = 12. The 5-12-13 triple.",
            "18 adds 13 and 5. Solving for a leg subtracts rather than adds, because the hypotenuse is the largest side.",
            "√194 comes from 13² + 5² = 194 — adding when you should subtract. That would give the hypotenuse of a triangle with legs 13 and 5, which is a different triangle.",
            "8 subtracts the sides directly, 13 − 5. The theorem operates on the squares, not the lengths."
          ],
          "hint": "The hypotenuse is already known, so you are solving for a leg. Which way does that rearrange the equation?"
        },
        {
          "id": "pyt-007",
          "topic": "Trigonometry",
          "difficulty": "medium",
          "stem": "In a right triangle, tan θ = 3/4 and the side opposite θ is 6. Find the adjacent side.",
          "choices": [
            "8",
            "4.5",
            "10",
            "3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. tan θ = opposite/adjacent = 3/4. With opposite = 6, solve 6/adjacent = 3/4, giving adjacent = 24/3 = 8. The triangle is a 6-8-10, the 3-4-5 scaled by 2.",
            "4.5 computes 6 × (3/4), applying the ratio in the wrong direction. Since tan θ < 1, the adjacent side must be LONGER than the opposite side.",
            "10 is the hypotenuse of this triangle, √(36 + 64). The question asks for the adjacent leg.",
            "3 repeats the numerator of the ratio. The ratio 3/4 is a proportion, not a set of lengths — the actual sides here are 6 and 8."
          ],
          "hint": "Set up the proportion 6/adjacent = 3/4 and cross-multiply. Sanity check: tan θ < 1 means adjacent > opposite."
        },
        {
          "id": "pyt-008",
          "topic": "Special Right Triangles",
          "difficulty": "medium",
          "stem": "In a 30-60-90 triangle the long leg is 5√3. Find the short leg and the hypotenuse.",
          "choices": [
            "short leg 5, hypotenuse 10",
            "short leg 5√3, hypotenuse 10√3",
            "short leg 15, hypotenuse 30",
            "short leg 5, hypotenuse 5√2"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The ratio is x : x√3 : 2x. The long leg is x√3 = 5√3, so x = 5, making the short leg 5 and the hypotenuse 2x = 10.",
            "This leaves the short leg equal to the long leg, which cannot happen in a 30-60-90 triangle — the sides opposite different angles have different lengths.",
            "15 multiplies by 3 instead of dividing by √3. Note that √3 · √3 = 3, so dividing 5√3 by √3 gives 5, not 15.",
            "5√2 is the 45-45-90 hypotenuse. In a 30-60-90 triangle the hypotenuse is exactly twice the short leg."
          ],
          "hint": "Divide the long leg by √3 to recover x, then double it for the hypotenuse."
        },
        {
          "id": "pyt-009",
          "topic": "Trigonometry",
          "difficulty": "medium",
          "stem": "From a point 50 ft from the base of a tower, the angle of elevation to the top is 40°. Find the tower's height. (tan 40° ≈ 0.839)",
          "choices": [
            "41.95 ft",
            "59.60 ft",
            "32.14 ft",
            "38.30 ft"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The 50 ft is the adjacent side and the height is opposite, so tan 40° = h/50, giving h = 50(0.839) ≈ 41.95 ft.",
            "59.60 ft divides by the tangent, 50/0.839. That would be right if the height were given and the distance sought.",
            "32.14 ft uses sin 40° ≈ 0.643. Sine relates the opposite side to the HYPOTENUSE, but 50 ft is the ground distance, which is adjacent.",
            "38.30 ft uses cos 40° ≈ 0.766, which relates the adjacent side to the hypotenuse — the wrong pairing again."
          ],
          "hint": "You have the ground distance (adjacent) and want the height (opposite). Which of the three ratios pairs exactly those two?"
        },
        {
          "id": "pyt-010",
          "topic": "Triangle Classification",
          "difficulty": "medium",
          "stem": "Classify the triangle with sides 6, 8 and 11 by its angles.",
          "choices": [
            "Right",
            "Acute",
            "Obtuse",
            "It cannot exist"
          ],
          "correctIndex": 2,
          "explanations": [
            "Right would require a² + b² = c². Here 36 + 64 = 100, which is not 121.",
            "Acute would require a² + b² > c². Here 100 < 121, so it fails in the other direction.",
            "Correct. Compare the sum of the squares of the two shorter sides with the square of the longest: 6² + 8² = 100 and 11² = 121. Since 100 < 121, the triangle is obtuse — the longest side is longer than a right angle would allow.",
            "It exists: 6 + 8 = 14 > 11, so the Triangle Inequality is satisfied. The classification test only applies once you have confirmed that."
          ],
          "hint": "Square the two shorter sides, add, and compare with the square of the longest. Less than means the angle opened past 90°.",
          "diagram": "<svg viewBox=\"0 0 240 134\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"25,105 215,105 78,42\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><text x=\"112\" y=\"124\" 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\">11</text><text x=\"38\" y=\"68\" 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\">6</text><text x=\"154\" y=\"66\" 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><text x=\"70\" y=\"60\" fill=\"#c0392b\" 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></svg>",
          "diagramCaption": "The angle opposite the longest side decides the classification. Not drawn to scale"
        },
        {
          "id": "pyt-011",
          "topic": "Special Right Triangles",
          "difficulty": "easy",
          "stem": "A 45-45-90 triangle has legs of length 7. Find the hypotenuse.",
          "choices": [
            "7√2",
            "14",
            "7√3",
            "7/√2"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The ratio x : x : x√2 gives a hypotenuse of 7√2 ≈ 9.9.",
            "14 doubles the leg. Doubling is the 30-60-90 relationship between the SHORT leg and the hypotenuse, not a 45-45-90 rule.",
            "7√3 uses the 30-60-90 constant. An isosceles right triangle carries √2.",
            "7/√2 divides instead of multiplying, giving a hypotenuse shorter than the legs — impossible, since the hypotenuse is always the longest side."
          ],
          "hint": "Both legs are 7, so Pythagoras gives c² = 49 + 49 = 98. Simplify √98."
        },
        {
          "id": "pyt-012",
          "topic": "Pythagorean Theorem",
          "difficulty": "hard",
          "stem": "Which set of three lengths is a Pythagorean triple?",
          "choices": [
            "9, 12, 16",
            "10, 24, 26",
            "5, 8, 13",
            "6, 7, 9"
          ],
          "correctIndex": 1,
          "explanations": [
            "9² + 12² = 81 + 144 = 225, but 16² = 256. Close, but not equal — 9, 12, 15 would have worked.",
            "Correct. 10² + 24² = 100 + 576 = 676, and 26² = 676. This is the 5-12-13 triple scaled by 2.",
            "5 + 8 = 13, so these three lengths do not even form a triangle — the two shorter sides collapse flat onto the longest. Certainly not a right one.",
            "6² + 7² = 36 + 49 = 85, but 9² = 81. Since 85 > 81, this triangle is acute rather than right."
          ],
          "hint": "Test a² + b² = c² on each, and watch for one that fails the Triangle Inequality before you even get that far."
        },
        {
          "id": "pyt-013",
          "topic": "Converse of Pythagoras",
          "difficulty": "medium",
          "stem": "A triangle has sides 6, 8 and 11. Classify it as right, acute, or obtuse.",
          "choices": [
            "Right",
            "Acute",
            "Obtuse",
            "No such triangle exists"
          ],
          "correctIndex": 2,
          "explanations": [
            "Right would require a² + b² = c² exactly. Here 6² + 8² = 100 while 11² = 121, so the equality fails.",
            "Acute requires a² + b² > c². Here 100 < 121, so the inequality runs the other way. The 6-8-10 triangle would be right; lengthening the longest side past 10 opens the angle beyond 90°.",
            "Correct. Compare 6² + 8² = 36 + 64 = 100 against 11² = 121. Since 100 < 121, the square of the longest side exceeds the sum of the other two squares, so the angle opposite it is obtuse.",
            "The triangle inequality is satisfied: 6 + 8 = 14 > 11, so this triangle certainly exists. It is simply obtuse."
          ],
          "hint": "Compare the sum of the squares of the two shorter sides against the square of the longest. The direction of the inequality names the triangle."
        },
        {
          "id": "pyt-014",
          "topic": "Geometric Mean",
          "difficulty": "hard",
          "stem": "In a right triangle, the altitude to the hypotenuse divides it into segments of length 4 and 9. Find the altitude.",
          "choices": [
            "6",
            "6.5",
            "13",
            "36"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The altitude to the hypotenuse is the geometric mean of the two segments it creates: h = √(p · q) = √(4 · 9) = √36 = 6.",
            "6.5 is the ARITHMETIC mean, (4 + 9)/2. The relationship here is geometric — a square root of a product, not an average of a sum.",
            "13 is the full hypotenuse, 4 + 9. That is a real length in the figure but not the altitude.",
            "36 is h², not h. The square root step was skipped."
          ],
          "hint": "The word 'mean' here is geometric, not arithmetic. That means a product under a radical."
        },
        {
          "id": "pyt-015",
          "topic": "30-60-90 Triangle",
          "difficulty": "medium",
          "stem": "A 30-60-90 triangle has a long leg of 5√3. Find the hypotenuse.",
          "choices": [
            "5",
            "10",
            "10√3",
            "5√6"
          ],
          "correctIndex": 1,
          "explanations": [
            "5 is the SHORT leg. In a 30-60-90 triangle the sides run x, x√3, 2x, so a long leg of 5√3 means x = 5 — but that is the short leg, not the hypotenuse.",
            "Correct. The ratio is x : x√3 : 2x. Matching the long leg x√3 = 5√3 gives x = 5, so the hypotenuse is 2x = 10.",
            "10√3 doubles the long leg. The hypotenuse is twice the SHORT leg, not twice the long one.",
            "5√6 comes from multiplying 5√3 by √2, which is the 45-45-90 relationship. That ratio belongs to the other special right triangle."
          ],
          "hint": "Write the three sides as multiples of x, match the one you were given, and solve for x before reading off the answer."
        },
        {
          "id": "pyt-016",
          "topic": "Angle of Elevation",
          "difficulty": "medium",
          "stem": "From a point 50 ft from the base of a tower, the angle of elevation to the top is 40°. Find the tower's height. (tan 40° ≈ 0.839)",
          "choices": [
            "41.95 ft",
            "59.6 ft",
            "32.1 ft",
            "65.3 ft"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The 50 ft is adjacent to the 40° angle and the height is opposite it, so tangent is the right ratio: tan 40° = height/50, giving height = 50(0.839) ≈ 41.95 ft.",
            "59.6 ft comes from dividing by the tangent instead of multiplying: 50/0.839. Dividing would be correct if the 50 ft were the opposite side rather than the adjacent one.",
            "32.1 ft comes from using sine instead of tangent. Sine needs the hypotenuse, and the 50 ft here is a horizontal leg on the ground, not the slanted line of sight.",
            "65.3 ft comes from using the cosine relationship inverted. Match the ratio to which two sides you actually have: opposite and adjacent means tangent."
          ],
          "hint": "Identify which sides the 50 ft and the unknown height are relative to the 40° angle, then pick the ratio that uses exactly those two."
        },
        {
          "id": "pyt-017",
          "topic": "Pythagorean Triples",
          "difficulty": "easy",
          "stem": "A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.",
          "choices": [
            "8",
            "12",
            "12.5",
            "√194"
          ],
          "correctIndex": 1,
          "explanations": [
            "8 comes from subtracting the given lengths, 13 − 5. The Pythagorean Theorem works with squares, so lengths cannot simply be subtracted.",
            "Correct. Rearranging a² + b² = c² gives b = √(c² − a²) = √(169 − 25) = √144 = 12. This is the 5-12-13 triple, worth memorizing alongside 3-4-5.",
            "12.5 is close to the right answer but comes from an approximation rather than the exact computation. √144 is exactly 12.",
            "√194 comes from ADDING the squares, 169 + 25. Adding is correct when solving for the hypotenuse; here the hypotenuse is already known, so you subtract."
          ],
          "hint": "The hypotenuse is given, so you are solving for a leg. That changes whether you add or subtract the squares."
        },
        {
          "id": "pyt-018",
          "topic": "Reasoning: When It Applies",
          "difficulty": "medium",
          "stem": "A student applies a² + b² = c² to a triangle with sides 5, 6 and 8 to find a missing angle. Why is this invalid?",
          "choices": [
            "The Pythagorean theorem only applies to RIGHT triangles, and 5² + 6² ≠ 8².",
            "The numbers are too small.",
            "The theorem only applies when all three sides are known.",
            "It is valid; every triangle satisfies a² + b² = c²."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The theorem holds only in right triangles. Here 25 + 36 = 61 while 64 = 8², so this triangle is not right and the theorem does not apply. The Law of Cosines is the general tool.",
            "Size is irrelevant. The theorem depends on the triangle having a right angle.",
            "Knowing all three sides is fine — it is the absence of a right angle that rules the theorem out.",
            "Only right triangles satisfy it. In general a² + b² is greater than c² for acute triangles and less for obtuse ones."
          ],
          "hint": "Test the three sides against the theorem before using it. Does the equation actually hold?",
          "kind": "reasoning"
        },
        {
          "id": "pyt-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "hard",
          "stem": "In a right triangle, a student sets up sin θ = 8/6, where 8 is a leg and 6 is the other leg. Without computing, how do you know this is wrong?",
          "choices": [
            "Sine is opposite over HYPOTENUSE, and a ratio of two legs can exceed 1 — which sine never does.",
            "Sine only works for angles under 45°.",
            "The student should have used 6/8 instead, which is a valid sine.",
            "Nothing is wrong."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Sine is opposite ÷ hypotenuse, and the hypotenuse is always the longest side, so sine of an acute angle is always between 0 and 1. 8/6 > 1 is impossible.",
            "Sine is defined for every angle; the 45° limit is invented.",
            "6/8 is a valid number, but it is still two LEGS — that ratio is the tangent, not the sine.",
            "A sine greater than 1 is impossible for an acute angle in a right triangle."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}