Geometry and proof · Ages 13–14

Pythagorean triples worksheet

Part of: Pythagoras' theorem

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Geometry and proof · Ages 13–14

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

1.3, 4, 5 is a Pythagorean triple, and so is 4 times it: , , .
Fill in each blank
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2.In the Pythagorean triple 6, y, 10, y is the middle one (6 < y < 10). What is y?
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3.5, 12 and one more number make a Pythagorean triple, the last the biggest of the three. What is it?
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4.For the odd number n = 15: (n² − 1) ÷ 2 = and (n² + 1) ÷ 2 = . The three make a Pythagorean triple.
Fill in each blank
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5.Make a Pythagorean triple with m = 7 and n = 2: m² − n² = , 2mn = , m² + n² = .
Fill in each blank
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6.A Pythagorean triple made from m and n (m > n) has m² − n² = 77 and m² + n² = 85. So m² = , n² = , m = and n = .
Fill in each blank
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7.In a Pythagorean triple the smallest number is the odd number 3, and the other two are 1 apart. What is the largest?
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8.In a Pythagorean triple the smallest number is odd and the other two are 1 apart, the largest being 61. What is the smallest?
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9.In a Pythagorean triple the smallest number is the odd number 25, and the other two are 1 apart. What is the largest?
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10.In a Pythagorean triple the smallest number is odd and the other two are 1 apart, the largest being 181. What is the smallest?
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13.Tsz-ching is making a right-angled triangle frame from three sticks, each a whole number of cm. The shortest is 3 cm and the other two differ by 1 cm. The other two are cm and cm.
Fill in each blank
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14.A program takes m = 7 and n = 3 and gives m² − n², 2mn and m² + n², that is , and . The three add up to .
Fill in each blank
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15.A right-angled triangle's legs are 56 cm and 192 cm. How long is the hypotenuse, in cm?
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16.Tsz-hin bends a 150 cm wire into a triangle whose sides are in the ratio 5 : 12 : 13. How long is the longest side, in cm?
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17.Check that 32, 60, 68 is a Pythagorean triple: put the steps in order.
- A. 32² + 60² = 1024 + 3600
- B. = 68²
- C. = 4624
Write the letters in the right order
Working:
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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?
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Show the answers and working

Pythagorean triples worksheet: Answers and working

1. 12, 16, 20
Times each by 4: (4 × 3)² + (4 × 4)² = 4² × (3² + 4²) = 4² × 5² = (4 × 5)².
2. 8
y² = 10² − 6² = 100 − 36 = 64, so y = 8.
3. 13
5² + 12² = 169 = 13², so it is 13.
4. 112, 113
15² = 225, (225 − 1) ÷ 2 = 112 and (225 + 1) ÷ 2 = 113; they are 1 apart, so 113² − 112² = 113 + 112 = 225 = 15².
5. 45, 28, 53
49 − 4 = 45, 2 × 7 × 2 = 28, 49 + 4 = 53; check: 45² + 28² = 53².
6. 81, 4, 9, 2
Add the two: 2m² = 77 + 85 = 162, so m² = 81; take one from the other: 2n² = 85 − 77 = 8, so n² = 4. So m = 9 and n = 2.
7. 5
Call the other two y and y + 1: (y + 1)² − y² = 2y + 1 = 3² = 9, so y = 4 and the largest is 5.
8. 11
Call the smallest n. By the odd-number rule 61 = (n² + 1) ÷ 2, so n² = 2 × 61 − 1 = 121 and n = 11; the other two are 60 and 61.
9. 313
Call the other two y and y + 1: (y + 1)² − y² = 2y + 1 = 25² = 625, so y = 312 and the largest is 313.
10. 19
Call the smallest n. By the odd-number rule 181 = (n² + 1) ÷ 2, so n² = 2 × 181 − 1 = 361 and n = 19; the other two are 180 and 181.

11. True
18² + 24² = 900 and 30² = 900: a triple only if they are equal.
12. D. 36, 48, 60
36² + 48² = 3600 = 60². 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. 4, 5
Sides 1 apart have squares differing by y + (y + 1) = 3² = 9: y = (9 − 1) ÷ 2 = 4, and the other 5.
14. 40, 42, 58, 140
40, 42 and 58, adding to 140; they are a Pythagorean triple: 40² + 42² = 58².
15. 200 cm
56 and 192 are 8 times 7 and 24, and 7, 24, 25 is a triple, so the hypotenuse is 25 × 8 = 200 cm, no big squares needed.
16. 65 cm
5 + 12 + 13 = 30 shares of 150 ÷ 30 = 5 cm, so the longest side is 13 × 5 = 65 cm (facing the right angle, 5, 12, 13 being a triple).
17. A → C → B
Square the two smaller numbers and add, 4624, then see whether it is the largest one squared: 68² = 4624, a triple.
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.

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