Question

Which set of side lengths forms a right triangle?
2, 3, 13
4, 6, 10
9, 12, 18
15, 36, 39

Answer 1

To determine which set of side lengths forms a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):

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

We need to check each set of lengths to see if this condition holds.

1. **Set: 2, 3, 13**
- Hypotenuse \( c = 13 \)
- \( 13^2 = 169 \)
- \( 2^2 + 3^2 = 4 + 9 = 13 \) (not equal to 169)

2. **Set: 4, 6, 10**
- Hypotenuse \( c = 10 \)
- \( 10^2 = 100 \)
- \( 4^2 + 6^2 = 16 + 36 = 52 \) (not equal to 100)

3. **Set: 9, 12, 18**
- Hypotenuse \( c = 18 \)
- \( 18^2 = 324 \)
- \( 9^2 + 12^2 = 81 + 144 = 225 \) (not equal to 324)

4. **Set: 15, 36, 39**
- Hypotenuse \( c = 39 \)
- \( 39^2 = 1521 \)
- \( 15^2 + 36^2 = 225 + 1296 = 1521 \) (equal to 1521)

The set **15, 36, 39** forms a right triangle, as it satisfies the Pythagorean theorem.