Question

Given the following side lengths, which triangle is a right triangle?

Group of answer choices

14, 20, 26

13, 14, 28

10, 14, 24

10, 24, 26

Answer 1

To determine which set of side lengths forms a right triangle, we can use the Pythagorean theorem. A triangle is a right triangle if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Let's check each group of side lengths:

1. **14, 20, 26**
- Longest side (hypotenuse) = 26
- Check if \( 26^2 = 14^2 + 20^2 \)
- \( 26^2 = 676 \)
- \( 14^2 + 20^2 = 196 + 400 = 596 \)
- \( 676 \neq 596 \) (not a right triangle)

2. **13, 14, 28**
- Longest side (hypotenuse) = 28
- Check if \( 28^2 = 13^2 + 14^2 \)
- \( 28^2 = 784 \)
- \( 13^2 + 14^2 = 169 + 196 = 365 \)
- \( 784 \neq 365 \) (not a right triangle)

3. **10, 14, 24**
- Longest side (hypotenuse) = 24
- Check if \( 24^2 = 10^2 + 14^2 \)
- \( 24^2 = 576 \)
- \( 10^2 + 14^2 = 100 + 196 = 296 \)
- \( 576 \neq 296 \) (not a right triangle)

4. **10, 24, 26**
- Longest side (hypotenuse) = 26
- Check if \( 26^2 = 10^2 + 24^2 \)
- \( 26^2 = 676 \)
- \( 10^2 + 24^2 = 100 + 576 = 676 \)
- \( 676 = 676 \) (this is a right triangle)

The only group of side lengths that forms a right triangle is **10, 24, 26**.