Geometry and proof · Ages 13–14

Pythagoras' theorem worksheet

Covers: Finding the hypotenuse, Finding a shorter side, The converse of Pythagoras' theorem, Using Pythagoras' theorem, Pythagorean triples

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Geometry and proof · Ages 13–14

Numsprite
numsprite.com

Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.A right-angled triangle has legs of 6 cm and 5 cm. How long is its hypotenuse, in cm, to 1 decimal place?
Answer:
2.A right-angled triangle's legs are 2 m and 45 cm. How long is its hypotenuse, in cm?
Answer:
3.A right-angled triangle's hypotenuse is 16 cm and one leg 15 cm. How long is the other leg, in cm, to 1 decimal place?
Answer:
4.In the right-angled triangle, the leg ? = cm.
Answer:
5.Two sides of a right-angled triangle are 9 cm and 13 cm, but it doesn't say whether either is the hypotenuse. If the 13 cm side is the hypotenuse, the third side = cm; if both are legs, the third side = cm.
Fill in each blank
Answer:
6.A triangle's sides are 10 cm, 24 cm and 26 cm. The longest is cm; the two shorter sides' squares add to , and the longest side's square is .
Fill in each blank
Answer:
7.A triangle with sides of 75 cm, 21 cm and 72 cm is right-angled. How long is the side opposite the right angle, in cm?
Answer:
8.In the right trapezium ABCD, AB = 4 cm, DC = 16 cm and AD = 16 cm. Find BC. The answer is cm
Answer:
9.In triangle ABC, D is on BC with AD ⊥ BC, AD = 12 cm, BD = 5 cm and DC = 35 cm. What is the perimeter of triangle ABC, in cm?
Answer:
10.For the odd number n = 5: (n² − 1) ÷ 2 = and (n² + 1) ÷ 2 = . The three make a Pythagorean triple.
Fill in each blank
Answer:

11.A right-angled triangle with legs of 16 cm and 30 cm has a hypotenuse of 34 cm.
True or false?
Answer:
12.Which of these is a Pythagorean triple?
- A. 18, 81, 83
- B. 18, 80, 82
- C. 18, 80, 83
- D. 19, 81, 83
Choose the right answer
Answer:

13.A flagpole is 14 m tall. A rope runs straight from its top to a point on the ground 8 m from its foot. About how long is the rope, in m, to 1 decimal place?
Working:
Answer:
14.A rectangular photo frame is 8 cm wide and 6 cm high. Find the length d of its diagonal: put the steps in order.
- A. = 64 + 36
- B. = 100
- C. d² = 8² + 6²
- D. d = 100 = 10
Write the letters in the right order
Working:
Answer:
15.A television's screen is 40 cm across the diagonal and 32 cm wide. The screen is cm high and its area is cm².
Fill in each blank
Working:
Answer:
16.A rope 360 m long is split into three pieces in the ratio 8 : 15 : 17 and pulled into a triangle. The pieces are m, m and m, and the triangle's area is m².
Fill in each blank
Working:
Answer:
17.A rectangular lawn has a diagonal of 82 m and one side of 80 m. How many m of railing go all the way round it?
Working:
Answer:
18.A right-angled triangle's sides are whole numbers of cm: the shortest is 6 cm and the other two differ by 2 cm. How long is the hypotenuse, in cm?
Working:
Answer:

Show the answers and working

Pythagoras' theorem worksheet: Answers and working

1. 7.8 cm
6² + 5² = 61, so the hypotenuse = 61 ≈ 7.8 cm.
2. 205 cm
Put both in one unit first: 2 m = 200 cm. 200² + 45² = 40000 + 2025 = 42025, so the hypotenuse = 42025 = 205 cm.
3. 5.6 cm
16² − 15² = 31, so the other leg = 31 ≈ 5.6 cm.
4. 39
8 cm is the hypotenuse: 8² − 5² = 64 − 25 = 39, not a square number, so ? = 39 cm.
5. 88, 250
With 13 cm the hypotenuse the third side is a leg, so subtract: 169 − 81 = 88; with both legs it is the hypotenuse, so add: 81 + 169 = 250. The 9 cm side is shorter than 13 cm, so it can't be the hypotenuse.
6. 26, 676, 676
The longest is 26 cm; the other two squared and added make 676, and 26² = 676. Only when the two are equal is the triangle right-angled.
7. 75 cm
The right angle faces the longest side: 21² + 72² = 5625 = 75², so it is 75 cm.
8. 20
Drop an upright from B to DC: a right triangle with legs 16 − 4 = 12 cm and the height 16 cm, so BC = 12² + 16² = 20 cm.
9. 90 cm
AB = 12² + 5² = 13 cm and AC = 12² + 35² = 37 cm; the perimeter = 13 + 37 + 5 + 35 = 90 cm.
10. 12, 13
5² = 25, (25 − 1) ÷ 2 = 12 and (25 + 1) ÷ 2 = 13; they are 1 apart, so 13² − 12² = 13 + 12 = 25 = 5².

11. True
16² + 30² = 256 + 900 = 1156, so the hypotenuse = 1156 = 34 cm.
12. B. 18, 80, 82
18² + 80² = 6724 = 82². Multiplying by one number keeps a triple; adding one doesn't.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 16.1 m
The pole stands at right angles to the ground and the rope is the hypotenuse: 14² + 8² = 260, so the rope = 260 ≈ 16.1 m.
14. C → A → B → D
The diagonal is the hypotenuse of the right triangle the sides make: square, add, then the root, d = 10 cm.
15. 24, 768
The diagonal is the hypotenuse: height = 40² − 32² = 24 cm; area = 32 × 24 = 768 cm².
16. 72, 135, 153, 4860
Each share is 360 ÷ 40 = 9 m, so 72, 135 and 153 m. 72² + 135² = 153², so by the converse of Pythagoras' theorem the two shorter sides are perpendicular: area = 72 × 135 ÷ 2 = 4860 m².
17. 196 m
The other side = 82² − 80² = 18 m; the perimeter = 2 × (80 + 18) = 196 m.
18. 10 cm
Call the others y and y + 2: (y + 2)² − y² = 4y + 4 = 6², so y + 1 = 9; the sides are 8 and 10, the hypotenuse 10 cm.

Numsprite · numsprite.com/en/worksheets/pythagoras-theorem