Question

Which of these triangles is a right triangle?

13,14,9
10,12,7
5,6,8
12,5,13

Answer 1

To determine which of these triangles is a right triangle, we can use the Pythagorean theorem. According to the theorem, a triangle is a right triangle if the squares of the lengths of the two shorter sides add up to the square of the longest side. That is, if \( a \), \( b \), and \( c \) are the lengths of the sides of a triangle where \( c \) is the longest side, then the triangle is a right triangle if \( a^2 + b^2 = c^2 \).

We will evaluate each set of sides:

1. **Triangle with sides 13, 14, 9**:
- Here, the longest side \( c = 14 \).
- Check: \( 9^2 + 13^2 = 81 + 169 = 250 \) and \( 14^2 = 196 \) (not a right triangle).

2. **Triangle with sides 10, 12, 7**:
- Here, the longest side \( c = 12 \).
- Check: \( 7^2 + 10^2 = 49 + 100 = 149 \) and \( 12^2 = 144 \) (not a right triangle).

3. **Triangle with sides 5, 6, 8**:
- Here, the longest side \( c = 8 \).
- Check: \( 5^2 + 6^2 = 25 + 36 = 61 \) and \( 8^2 = 64 \) (not a right triangle).

4. **Triangle with sides 12, 5, 13**:
- Here, the longest side \( c = 13 \).
- Check: \( 5^2 + 12^2 = 25 + 144 = 169 \) and \( 13^2 = 169 \) (it is a right triangle).

Based on the checks, the triangle with sides **12, 5, and 13** is a right triangle.