Question

Which set of numbers could represent the lengths of the sides of a right triangle?

8, 15, 17
10, 15, 20
12, 18, 22
7,9, 11

Answer 1

The first set: 8, 15, and 17.

Explanation:

Pair: 8, 15, 17

By the pythagorean theorem, a triangle is a right triangle if and only if

`\text{longest side}^2 = \text{first shorter side}^2 + \text{second shorter side}^2`.

In this case,

`\text{longest side}^2 = 17^2 = 289`.

`\begin{aligned}&\text{first shortest side}^2 + \text{second shortest side}^2 \\ &= 8^2 + 15^2\\ &=64 + 225 = 289 \end{aligned}`.

In other words, indeed
`\text{hypotenuse}^2 = \text{first leg}^2 + \text{second leg}^2`. Hence, 8, 15, 17 does form a right triangle.

Similarly, check the other pairs. Keep in mind that the square of the longest side should be equal to the sum of the square of the two

Pair: 10, 15, 20

Factor out the common factor
`2` to simplify the calculations.

`\text{longest side}^2 = 20^2 = 400`

`\begin{aligned}&\text{first shortest side}^2 + \text{second shortest side}^2 \\ &= 10^2 + 15^2\\ &=100 + 225 = 325 \end{aligned}`.

`\text{longest side}^2 \\e \text{first shorter side}^2 + \text{second shorter side}^2`.

Hence, by the pythagorean theorem, these three sides don't form a right triangle.

Pair: 12, 18, 22

`\text{longest side}^2 = (2* 11)^2 = 2^2 * 121`.

`\begin{aligned}&\text{first shortest side}^2 + \text{second shortest side}^2 \\ &= (2 * 6)^2 + (2 * 9)^2\\ &=2^2 *(36 + 81) = 2^2 * 117 \end{aligned}`.

`\text{longest side}^2 \\e \text{first shorter side}^2 + \text{second shorter side}^2`.

Hence, by the pythagorean theorem, these three sides don't form a right triangle.

Pair: 7, 9, 11

`\text{longest side}^2 = 11^2 = 121`.

`\begin{aligned}&\text{first shortest side}^2 + \text{second shortest side}^2 \\ &= 7^2 + 9^2\\ &=49+ 81 = 130 \end{aligned}`.

`\text{longest side}^2 \\e \text{first shorter side}^2 + \text{second shorter side}^2`.

Hence, by the pythagorean theorem, these three sides don't form a right triangle.