NSW 8 Pythagoras' Theorem worked solutions

NSW8

8.01 Pythagoras’ Theorem Worked Solutions

Pythagoras’ theorem 8.01

Understanding

b. 122 + 352 = 144 + 1225 = 1369

1. a2 + b2 = c2

372 = 37 × 37

c is the hypotenuse.

= 1369

2. a. Hypotenuse is the longest side. ∴ 25

→ 122 + 352 = 372

b. Two remaining sides. 7 and 24

∴ Yes

c. 92 + 162 = 81 + 256

3. Hypotenuse is the longest side and also the side opposite of the right angle.

= 337 2

18 = 18 × 18

a. Z

= 324

b. M

→ 92 + 162 = ̸ 182

c. Y

∴ No

d. C

d. 12 + 12 = 1 + 1 =2 √ √ ( 2) = 2 × 2

Fluency

√

4. a. 32 + 42 = 9 + 16

2

=2 √ → 12 + 12 = ( 2)2 ∴ Yes

= 25 b. 52 = 5 × 5 = 25

6. Equation: a2 + b2 = c2

c. Yes because 32 + 42 = 52

a. Hypotenuse: c ∴ 122 + 92 = c2

5. a. 172 + 102 = 289 + 100 = 389

b. Hypotenuse: 26 ∴ 102 + b2 = 262

2

23 = 23 × 23 = 529

c. Hypotenuse: 19 ∴ x2 + 132 = 192

→ 172 + 102 ̸= 232

∴ No

d. Hypotenuse: c ∴ 9 2 + 32 = c 2

1

c. 282 + x2 = 532

7. a. 62 + 82 = 36 + 64

784 + x2 = 2809

= 100 132 = 13 × 13

x2 = 2025

= 169 → 100 ̸= 169 ∴ No

x = 45 d.

142 + 482 = x2 196 + 2304 = x2

b. 32 + 42 = 9 + 16

x2 = 2500

= 25

x = 50

52 = 5 × 5

9. a. 82 + 132 = 64 + 169

= 25 → 32 + 42 = 52 ∴ Yes 2

= 233 2

19 = 19 × 19

2

= 361 → 233 ̸= 361

c. 2 + 4 = 4 + 16 = 20

∴ No

62 = 6 × 6

b. 92 + 162 = 81 + 256

= 36 → 20 ̸= 36 ∴ No 2

= 337 2

25 = 25 × 25

2

= 625 → 337 ̸= 625

d. 5 + 12 = 25 + 144 = 169

∴ No

132 = 13 × 13

c. 202 + 212 = 400 + 441

= 169 → 52 + 122 = 132

= 841

∴ Yes

2

29 = 841 → 202 + 212 = 292

8. Last number is hypotenuse.

2

2

∴ Yes 2 2 1 1 1 1 + = + d.

3 4 9 16 9 16 + = 144 144 25 = 144 2 1 1 = 5 25 1 25 ̸= → 144 25 ∴ No

2

a. x + 15 = 17

x2 + 225 = 289 x2 = 64 x=8 2

2

b. 7 + x = 252 49 + x2 = 625 x2 = 576 √ x = 576 x = 24 2

10. a.

Problem-solving

x2 + 962 = 1002 2

x + 9216 = 10 000

14. Right-angled triangle means the opposite side of the right angle (hypotenuse) is the longest side.

x2 = 784 √ x = 784 x = 28 b.

2

2

∴ Florey and Giralang

22 + 120 = x

2

484 + 14 400 = x

2

15. a. 202 + 482 = 400 + 2304 = 2704

x2 = 14 884 √ x = 14 884

522 = 52 × 52 = 2704

x = 122

→ 202 + 482 = 522 ∴ Yes

c. x2 + 82 = 102

b. 20 + 48 + 52 = 120 m

x2 + 64 = 100 x2 = 36 √ x = 36

c. i. 302 + 402 = 900 + 1600

x=6

2

= 2500 50 = 50 × 50 = 2500

d. 82 + x2 = 172

→ 302 + 402 = 502

64 + x2 = 289

∴ Yes ii. 262 + 412 = 676 + 1681

2

x = 225 √ x = 225

= 2357 2

53 = 53 × 53

x = 15 e.

= 2809 → 2357 = ̸ 2809 ∴ No iii. 142 + 482 = 196 + 2304

x2 + 402 = 412 x2 + 1600 = 1681 x2 = 81 √ x = 81

= 2500 2

50 = 50 × 50

x=9

= 2500 → 142 + 482 = 502 ∴ Yes iv. 242 + 452 = 576 + 2025

Reasoning 11. Refer to answer key.

= 2601 2

51 = 51 × 51 12. Refer to answer key.

= 2601 → 242 + 452 = 512 ∴ Yes

13. Refer to answer key.

3

16.

20

b a

a2 + b2 = 202 a2 + b2 = 400 Using trial and error: a and b = 12 and 16

17.

65

b a

a2 + b2 = 652 a2 + b2 = 4225 Using trial and error: a and b = 39, 52 or 25, 60

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