Question

Which side lengths form a right triangle?
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[5, 6, 7\]
A
\[5, 6, 7\]
(Choice B)
\[6, 8, 10\]
B
\[6, 8, 10\]
(Choice C)
\[9, 9, 18\]
C
\[9, 9, 18\]

Answer 1

To determine whether a set of side lengths can form a right triangle, we can use the Pythagorean theorem, which states that in a right triangle with sides \(a\), \(b\), and hypotenuse \(c\) (where \(c\) is the longest side), the following equation must hold true:

\[
a^2 + b^2 = c^2
\]

Now, let's evaluate each choice:

**(Choice A) \(5, 6, 7\)**:
- Here, the largest side (hypotenuse) is 7. We check if:
\[
5^2 + 6^2 = 25 + 36 = 61
\]
\[
7^2 = 49
\]
Since \(61 \neq 49\), these lengths do not form a right triangle.

**(Choice B) \(6, 8, 10\)**:
- The hypotenuse is 10. We check if:
\[
6^2 + 8^2 = 36 + 64 = 100
\]
\[
10^2 = 100
\]
Since \(100 = 100\), these lengths do form a right triangle.

**(Choice C) \(9, 9, 18\)**:
- The hypotenuse is 18. We check if:
\[
9^2 + 9^2 = 81 + 81 = 162
\]
\[
18^2 = 324
\]
Since \(162 \neq 324\), these lengths do not form a right triangle.

Therefore, the only valid choice that forms a right triangle is:

**(Choice B) \(6, 8, 10\)**.