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

To determine the type of triangle with sides of lengths 5, 12, and 13, one effective way is to apply the Pythagorean theorem, which states that for a triangle to be classified as a right triangle, the square of the length of the longest side must equal the sum of the squares of the lengths of the other two sides.

In this case, the longest side is 13. We calculate the squares of each side:

• The square of 5 is (5^2 = 25).

• The square of 12 is (12^2 = 144).

• The square of 13 is (13^2 = 169).

Next, we add the squares of the two shorter sides together:

(5^2 + 12^2 = 25 + 144 = 169).

Now we compare this sum to the square of the longest side:

Since (169 = 169), this confirms that the sides meet the Pythagorean theorem condition, indicating that the triangle is indeed a right triangle.

The additional classifications of triangles—such as equilateral, isosceles, or scalene—do not apply here. An equilateral triangle has all sides