The problem asks to determine if a triangle with sides 71, 52, and 40 is right, acute, or obtuse, using the Converse of the Pythagorean Theorem.

1. Problem Description

The problem asks to determine if a triangle with sides 71, 52, and 40 is right, acute, or obtuse, using the Converse of the Pythagorean Theorem.

2. Solution Steps

The Converse of the Pythagorean Theorem states:

- If $c^2 = a^2 + b^2$, then the triangle is a right triangle.

- If $c^2 > a^2 + b^2$, then the triangle is an obtuse triangle.

- If $c^2 < a^2 + b^2$, then the triangle is an acute triangle.

Here, $a = 40$, $b = 52$, and $c = 71$, where $c$ is the longest side.

Calculate $c^2$:

$c^2 = 71^2 = 5041$

Calculate $a^2 + b^2$:

$a^2 + b^2 = 40^2 + 52^2 = 1600 + 2704 = 4304$

Since $5041 > 4304$, we have $c^2 > a^2 + b^2$.

Therefore, the triangle is obtuse. The triangle is obtuse because the square of the largest side is greater than the sum of the squares of the other two sides.

3. Final Answer

The triangle is obtuse because the square of the largest side is greater than the sum of the squares of the other two sides.