The problem states that the sides of a triangle are 52, 42, and 27. We need to use a modified Pythagorean theorem to determine if the triangle is right, acute, or obtuse.

1. Problem Description

The problem states that the sides of a triangle are 52, 42, and

2

7. We need to use a modified Pythagorean theorem to determine if the triangle is right, acute, or obtuse.

2. Solution Steps

First, we identify the longest side, which is

5

2. Then we calculate the square of the longest side:

$52^2 = 2704$

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

$42^2 = 1764$

$27^2 = 729$

$42^2 + 27^2 = 1764 + 729 = 2493$

Now we compare the square of the longest side with the sum of the squares of the other two sides:

$52^2 = 2704$

$42^2 + 27^2 = 2493$

Since $52^2 > 42^2 + 27^2$, the triangle is obtuse.

In general:

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

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

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

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.