Finding the Hypotenuse — Solutions

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1. Q1 — Find the hypotenuse (integer legs)

   1. (a) a=3, b=4: c²=9+16=25 → c=5 cm
   2. (b) a=5, b=12: c²=25+144=169 → c=13 cm
   3. (c) a=6, b=8: c²=36+64=100 → c=10 cm
   4. (d) a=8, b=15: c²=64+225=289 → c=17 cm
   5. (e) a=7, b=24: c²=49+576=625 → c=25 cm
   6. (f) a=9, b=40: c²=81+1600=1681 → c=41 cm
   7. (g) a=4, b=7: c²=16+49=65 → c=√65 ≈ 8.06 cm
   8. (h) a=5, b=9: c²=25+81=106 → c=√106 ≈ 10.30 cm
   9. (i) a=6, b=6: c²=72 → c=6√2 ≈ 8.49 cm
   10. (j) a=10, b=10: c²=200 → c=10√2 ≈ 14.14 cm

2. Q2 — Find the hypotenuse (decimal legs)

   1. (a) 2.5, 3.5: c=√(6.25+12.25)=√18.5 ≈ 4.30 m
   2. (b) 1.5, 2.0: c=√(2.25+4.00)=√6.25=2.50 m
   3. (c) 4.2, 5.6: c=√(17.64+31.36)=√49=7.00 cm
   4. (d) 3.6, 4.8: c=√(12.96+23.04)=√36=6.00 m
   5. (e) 7.5, 10.0: c=√(56.25+100)=√156.25=12.50 cm
   6. (f) 0.6, 0.8: c=√(0.36+0.64)=√1=1.00 m
   7. (g) 5.5, 7.2: c=√(30.25+51.84)=√82.09 ≈ 9.06 m
   8. (h) 9.1, 12.3: c=√(82.81+151.29)=√234.10 ≈ 15.30 cm

3. Q3 — Exact surd answers

   1. (a) 1, 1: c=√(1+1)=√2 ≈ 1.41 cm
   2. (b) 2, 3: c=√(4+9)=√13 ≈ 3.61 cm
   3. (c) 4, 6: c=√(16+36)=√52=2√13 ≈ 7.21 cm
   4. (d) 5, 5: c=√(25+25)=√50=5√2 ≈ 7.07 cm
   5. (e) 3, 7: c=√(9+49)=√58 ≈ 7.62 cm
   6. (f) 6, 10: c=√(36+100)=√136=2√34 ≈ 11.66 cm

4. Q4 — True or False

   1. True. The hypotenuse is opposite the right angle and is always the longest side.
   2. False. You cannot add the sides — you must add their squares. c²=36+64=100, so c=10, not 14.
   3. True. Pythagoras’ theorem applies only to right-angled triangles.
   4. True. 5²+12²=25+144=169=13². ✓
   5. True. In any triangle, the largest angle is opposite the longest side. In a right triangle, 90° is the largest angle, so the hypotenuse is opposite it.
   6. True. If legs become 2a and 2b, then c′ = √(4a²+4b²) = 2√(a²+b²) = 2c.

5. Q5 — Ladder against a wall

   1. The triangle has a vertical leg (wall height = unknown), horizontal leg (3 m), and hypotenuse (ladder = 5 m).
   2. h² = 5² − 3² = 25 − 9 = 16 → h = 4 m
   3. h² = 25 − 16 = 9 → h = 3 m. The ladder only reaches 3 m, which is lower (by 1 m) when the base is moved out further.

6. Q6 — Diagonal of a rectangle

   1. The diagonal divides the rectangle into two right triangles with legs 9 m and 12 m.
   2. c²=81+144=225 → c=15 m (exact)
   3. Cost = 15 × $18.50 = $277.50

7. Q7 — Screen diagonal

   1. c=√(80²+45²)=√(6400+2025)=√8425≈91.79≈92 cm
   2. Width²=55²−27²=3025−729=2296 → Width=√2296≈47.9≈48 inches

8. Q8 — Pythagorean triples check

   1. (a) 6,8,10: 36+64=100=10² ✓ Yes
   2. (b) 5,12,13: 25+144=169=13² ✓ Yes
   3. (c) 4,5,6: 16+25=41 ≠ 36 ✗ No
   4. (d) 8,15,17: 64+225=289=17² ✓ Yes
   5. (e) 9,40,41: 81+1600=1681=41² ✓ Yes
   6. (f) 7,24,26: 49+576=625 ≠ 676=26² ✗ No (the hypotenuse should be 25)

9. Q9 — 3D diagonal of a box

   1. Floor diagonal: d_floor = √(8²+6²) = √(64+36) = √100 = 10 m
   2. Space diagonal: d = √(10²+3²) = √(100+9) = √109 ≈ 10.44 m
   3. 10.44 m < 10.5 m, so yes, the pole will fit (just barely).
   4. Rounding √109 ≈ 10.44 down to nearest whole number: the longest whole-number pole that fits is 10 m.

10. Q10 — Multi-step: antenna cables

    1. Cable A: c=√(12²+5²)=√(144+25)=√169=13 m (exact)
    2. Cable B: c=√(12²+9²)=√(144+81)=√225=15 m (exact)
    3. Total cable = 13 + 15 = 28 m. Cost = 28 × $12.40 = $347.20
    4. Half-height cable: c=√(6²+4²)=√(36+16)=√52=2√13≈7.21 m