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Middle Years

8.02 Pythagorean triples

Worksheet
Pythagorean triples
1

Determine whether the following are Pythagorean triples:

a

\left(5, 12, 13\right)

b

\left(12, 5, 13\right)

c

\left(8, 14, 18\right)

d

\left(9, 16, 25\right)

e

\left(300, 400, 500\right)

f

\left(9, 12, 15\right)

g

\left(15, 8, 17\right)

h

\left(\dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}\right)

2

Complete the following Pythagorean triples:

a

⬚, 15, 17

b

7, ⬚, 25

c

28, ⬚, 53

d

14, 48, ⬚

3

Peter knows the two smallest numbers in a Pythagorean triple, which are 3 and 4. Find the number Peter needs to complete the triple.

4

Sean knows the two largest numbers in a Pythagorean triple, which are 41 and 40. Find the number Sean needs to complete the triple.

5

Given that the length of the hypotenuse of a right-angled triangle is 20, find the two other side lengths that would complete a Pythagorean triple.

6

Find the number that completes a Pythagorean triple given the following:

a

The two largest numbers are 100 and 96.

b

The two smallest numbers are 22 and 120.

7

If x, y, and z form a Pythagorean triple, will 3 x, 3 y and 3 z be a Pythagorean triple?

Right angled triangles
8

Determine whether the following triples of numbers represent the sides of right-angled triangles:

a

\left(2, 4, 6\right)

b

\left(5, 12, 13\right)

c

\left(6, 8, 13\right)

d

\left(3, 4, 5\right)

9

Identify the hypotenuse of the following triangles:

a
b
c
10

For each of the following triangles, let a and b represent the two shorter sides and c the length of the longest side:

i

Find a^{2} + b^{2}.

ii

Find c^{2}.

iii

Is the triangle a right-angled triangle?

a
b
11

Consider the following right-angled triangle:

Write the equation that can be constructed from the given information.

12

A right-angled triangle has side lengths x, y, and z. Will a triangle with side lengths of 5 x, 5 y and 5 z make a right-angled triangle as well?

13

Consider the right-angled triangle with sides \left(8, 15, 17\right).

a

Find the length of the side that is opposite the largest angle.

b

Find the lengths of the two sides that are next to the right angle.

14

Consider a right-angled triangle with shorter side lengths of 15 units and 36 units.

a

Find the length of the hypotenuse in the triangle.

b

The triple is a multiple of a common Pythagorean triple. Which one?

15

Consider a right-angled triangle with side length of 40 \text{ cm} and hypotenuse of 85 \text{ cm}.

a

Find the length of the unknown side in the triangle.

b

The triple is a multiple of a common Pythagorean triple. Which one?

Applications
16

Three towns Melba, Florey and Giralang are positioned as shown in the diagram:

Which two towns are furthest apart, assuming a direct route is taken?

17

A group of engineering students have made a triangle out of some wooden strips. They have made a triangle with sides lengths 20 \text{ m}, 48 \text{ m}, and 52 \text{ m}.

a

Is the triangle they make a right-angled triangle?

b

How many metres of wooden strips did they use to make the triangle?

c

Find the other right-angled triangles they could create using the exact same length of wooden strips.

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