To determine if each 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 (the hypotenuse) is equal to the sum of the squares of the other two sides.
Let's evaluate each set of side lengths:
-
9, 13, 15:
Longest side = 15
\( 15^2 = 225 \)
\( 9^2 + 13^2 = 81 + 169 = 250 \)
\( 225 \neq 250 \) → Not a right triangle. -
6, 8, 12:
Longest side = 12
\( 12^2 = 144 \)
\( 6^2 + 8^2 = 36 + 64 = 100 \)
\( 144 \neq 100 \) → Not a right triangle. -
5, 9, 11:
Longest side = 11
\( 11^2 = 121 \)
\( 5^2 + 9^2 = 25 + 81 = 106 \)
\( 121 \neq 106 \) → Not a right triangle. -
12, 16, 20:
Longest side = 20
\( 20^2 = 400 \)
\( 12^2 + 16^2 = 144 + 256 = 400 \)
\( 400 = 400 \) → This is a right triangle.
Conclusion: The only set of side lengths that forms a right triangle is 12, 16, 20.