If a triangle has sides of lengths 5, 12, and 13, is it a right triangle?

To determine whether a triangle with side lengths of 5, 12, and 13 is a right triangle, one can apply the Pythagorean theorem. This theorem states that in any right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the side lengths are 5, 12, and 13. Here, 13 is the longest side, suggesting it may be the hypotenuse. We can perform the calculation as follows:

1. Calculate the square of the longest side: ( 13^2 = 169 ).

2. Calculate the sum of the squares of the other two sides: ( 5^2 + 12^2 = 25 + 144 = 169 ).

Since 169 (the square of the hypotenuse) equals 169 (the sum of the squares of the other two sides), this confirms that the triangle with side lengths 5, 12, and 13 satisfies the conditions of the Pythagorean theorem. Therefore, it is indeed a right triangle.

This conclusion rests solidly on the principles of geometry and the specific