The problem asks us to determine if a triangle with sides 77, 82, and 15 is a right, acute, or obtuse triangle, using the Pythagorean theorem. We are also asked to justify the answer by comparing the square of the longest side with the sum of the squares of the other two sides.

1. Problem Description

The problem asks us to determine if a triangle with sides 77, 82, and 15 is a right, acute, or obtuse triangle, using the Pythagorean theorem. We are also asked to justify the answer by comparing the square of the longest side with the sum of the squares of the other two sides.

2. Solution Steps

First, we identify the longest side, which is

8

2. Next, we calculate the square of the longest side:

$82^2 = 6724$

Then, we calculate the sum of the squares of the other two sides:

$77^2 + 15^2 = 5929 + 225 = 6154$

Now, we compare the square of the longest side ($82^2$) to the sum of the squares of the other two sides ($77^2 + 15^2$).

$6724 > 6154$

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

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

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

Since $82^2 > 77^2 + 15^2$, the triangle is an obtuse triangle.

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.