Topic Review — Pythagoras’ Theorem — Solutions

Click any answer to watch the solution video.

1. Finding the hypotenuse

   1. Legs 9, 12: c=√(81+144)=√225=15 cm
   2. Legs 5, 12: c=√(25+144)=√169=13 m
   3. Legs 8, 15: c=√(64+225)=√289=17 cm
   4. Legs 7, 24: c=√(49+576)=√625=25 cm
   5. Legs 6, 8: c=√(36+64)=√100=10 m

2. Finding a shorter side

   1. c=13, b=5: a=√(169−25)=√144=12 cm
   2. c=25, b=20: a=√(625−400)=√225=15 m
   3. c=17, b=15: a=√(289−225)=√64=8 cm
   4. c=10, b=√19: a=√(100−19)=√81=9 m
   5. c=29, b=20: a=√(841−400)=√441=21 cm

3. Converse of Pythagoras

   1. 10, 24, 26: 10²+24²=100+576=676=26². Right-angled. ✓
   2. 6, 9, 12: 6²+9²=36+81=117≠144=12². Not right-angled.
   3. 15, 36, 39: 15²+36²=225+1296=1521=39². Right-angled. ✓ (3–12–13 ×3)
   4. 7, 7, 10: 7²+7²=98 < 100=10². Not right-angled; obtuse (a²+b² < c²).

4. Mixed applications

   1. Block 12×5×4: d_base=√(144+25)=13. d=√(169+16)=√185≈13.60 cm
   2. A(−2,1) to B(4,9): d=√(36+64)=√100=10
   3. Isosceles base=16, sides=17: h=√(289−64)=√225=15 cm
   4. Pole 24m, peg 10m: cable=√(576+100)=√676=26 m
   5. c=2√10, b=2: a=√(40−4)=√36=6 cm

5. Problem solving

   1. Roof: legs 3.6m and 4.8m: Rafter=√(12.96+23.04)=√36=6 m. Area=(1/2)(3.6)(4.8)=8.64 m²
   2. Helicopter 40km east, 30km south: Distance=√(1600+900)=√2500=50 km. Time=50/120 hrs = 5/12 hrs = 25 minutes.
   3. Legs in ratio 3:4, hypotenuse 20cm:
      1. Let legs = 3k and 4k. (3k)²+(4k)²=20² ⇒ 9k²+16k²=400 ⇒ 25k²=400
      2. k²=16 ⇒ k=4. Legs = 12 cm and 16 cm.
      3. 12²+16²=144+256=400=20². ✓ This is a 3–4–5 triple ×4.
   4. Laptop screen 15-inch diagonal, aspect ratio 16:9:
      1. Let w=16k, h=9k. (16k)²+(9k)²=15² ⇒ 256k²+81k²=225 ⇒ 337k²=225 ⇒ k=√(225/337)=15/√337. Width=240/√337≈13.08 inches. Height=135/√337≈7.36 inches.
      2. Area = 13.08 × 7.36 ≈ 96.3 in²