Geometry and proof · Ages 13–14
Finding a shorter side worksheet
Part of: Pythagoras' theorem
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Finding a shorter side worksheet
Geometry and proof · Ages 13–14
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.In the right-angled triangle, how long is the leg ?, in cm?
Answer:
2.A right-angled triangle's hypotenuse is 13 cm and one leg 5 cm: 132 − 52 = − = , so the other leg = cm.
Fill in each blank
Answer:
3.A leg by the difference of two squares: hypotenuse 51 cm, leg 45 cm, 512 − 452 = (51 + 45)(51 − 45) = × = , so the other leg = cm.
Fill in each blank
Answer:
4.A right-angled triangle's hypotenuse is 16 cm and one leg 10 cm. How long is the other leg, in cm, to 1 decimal place?
Answer:
5.In the right-angled triangle, the leg ? = cm.
Answer:
6.A right-angled triangle's hypotenuse is 11 cm and one leg 4 cm. How long is the other leg, in cm, to 1 decimal place?
Answer:
7.The picture shows an isosceles right-angled triangle with a hypotenuse of 6 cm. Each leg = cm.
Answer:
8.In a right-angled triangle the hypotenuse c is 18 cm longer than the leg x, and the other leg is 30 cm. 302 = c2 − x2 = (c + x)(c − x), so c + x = ÷ 18 = and x = cm.
Fill in each blank
Answer:
9.Two sides of a right-angled triangle are 11 cm and 12 cm, but it doesn't say whether either is the hypotenuse. If the 12 cm side is the hypotenuse, the third side = cm; if both are legs, the third side = cm.
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
10.The right-angled triangle's hypotenuse is 39 cm and one leg 15 cm. How long is the leg ?, in cm?
- A. 36
- B. 24
- C. 1296
- D. 54
Choose the right answer
Answer:
11.A right-angled triangle has a hypotenuse of 20 cm, one leg of 16 cm and the other leg 12 cm.
True or false?
Answer:
12.A right-angled triangle has a hypotenuse of 68 cm, one leg of 60 cm and the other leg 33 cm.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A 6.5 m ladder leans on a wall with its top 6 m up. The top slides 0.4 m down the wall. How far does the foot move out, in m?
Working:
Answer:
14.Ka-lok is flying a kite on 68 m of string pulled straight. The kite is directly above a point 32 m in front. Leaving out the height of the hand, how high is the kite, in m?
Working:
Answer:
15.A television's screen is 125 cm across the diagonal and 100 cm wide. The screen is cm high and its area is cm².
Fill in each blank
Working:
Answer:
16.A playground slide is 7 m long and its top is 4 m above the ground. About how far is the bottom of the slide from the point right under its top, in m, to 1 decimal place?
Working:
Answer:
17.A 4 m ladder leans against a wall, its foot 2.4 m from the wall. 42 − 2.42 = , so the top is m above the ground.
Fill in each blank
Working:
Answer:
18.A 58 m ladder leans on a wall with its foot 40 m out. Taking the top to be h m up, find h: put the steps in order.
- A. h2 + 402 = 582
- B. h = 42
- C. h2 = 1764
- D. h2 = 3364 − 1600
Write the letters in the right order
Working:
Answer:
Show the answers and working
Finding a shorter side worksheet: Answers and working
Part A: Working out and filling in
1. 10 cm
26 cm is the hypotenuse: 262 − 242 = 676 − 576 = 100, so ? = 100 = 10 cm.
2. 169, 25, 144, 12
A leg is a subtraction: 169 − 25 = 144, and 144 = 12 × 12, so the other leg is 12 cm.
3. 96, 6, 576, 24
(51 + 45)(51 − 45) = 96 × 6 = 576, the same as 2601 − 2025; 576 = 24.
4. 12.5 cm
162 − 102 = 156, so the other leg = 156 ≈ 12.5 cm.
5. 40
7 cm is the hypotenuse: 72 − 32 = 49 − 9 = 40, not a square number, so ? = 40 cm.
6. 10.2 cm
112 − 42 = 105, so the other leg = 105 ≈ 10.2 cm.
7. 18
The legs are equal, say x: x2 + x2 = 62, 2x2 = 36, x2 = 18, so x = 18.
8. 900, 50, 16
c − x = 18, so c + x = 900 ÷ 18 = 50. Take one from the other: 2x = 50 − 18, so x = 16 cm (and the hypotenuse c = 34 cm).
9. 23, 265
With 12 cm the hypotenuse the third side is a leg, so subtract: 144 − 121 = 23; with both legs it is the hypotenuse, so add: 121 + 144 = 265. The 11 cm side is shorter than 12 cm, so it can't be the hypotenuse.
Part B: Choose, or true or false
10. A. 36
392 − 152 = 1521 − 225 = 1296, so ? = 1296 = 36 cm.
11. True
202 − 162 = 400 − 256 = 144, so the other leg = 144 = 12 cm.
12. False
682 − 602 = 4624 − 3600 = 1024, so the other leg = 1024 = 32 cm.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 0.8 m
Before, the foot is 6.52 − 62 = 2.5 m out; after, the top is 5.6 m up and the foot 6.52 − 5.62 = 3.3 m out; it moves 3.3 − 2.5 = 0.8 m.
14. 60 m
The string is the hypotenuse: 682 − 322 = 4624 − 1024 = 3600, so the height = 3600 = 60 m.
15. 75, 7500
The diagonal is the hypotenuse: height = 1252 − 1002 = 75 cm; area = 100 × 75 = 7500 cm².
16. 5.7 m
The slide is the hypotenuse: 72 − 42 = 33, so the distance = 33 ≈ 5.7 m.
17. 10.24, 3.2
The ladder is the hypotenuse and the height a leg: 42 − 2.42 = 10.24, so the height = 10.24 = 3.2 m.
18. A → D → C → B
Pythagoras' theorem gives the equation; rearranged, h2 = 1764, and the root gives h = 42.
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