A triangle has side lengths 10, 15, and 7. Is the triangle acute, obtuse, or right? Show your work and explain.

I think it's gonna be an obtuse and maybe right....might be wrong...

a = 7, b = 10, c = 15.

a^2 + b^2 = c^2.
49 + 100 = 149. But c^2 = 225. So it is
NOT a rt. triangle.

To determine whether a triangle is acute, obtuse, or right, we need to use the Pythagorean theorem and compare the squares of the triangle's side lengths.

The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side. So, let's check if the given side lengths of 10, 15, and 7 satisfy this condition.

First, we want to find the longest side of the triangle. Comparing the three side lengths, 15 is the longest.

Now, let's calculate the squares of the other two sides:
- The square of the first side, 10, is 10^2 = 100.
- The square of the second side, 7, is 7^2 = 49.

Next, we'll add these two squares to check if their sum is equal to the square of the longest side:
100 + 49 = 149.

Since the sum of the squares of the two shorter sides (100 and 49) is less than the square of the longest side (15^2 = 225), we can conclude that this triangle is obtuse.

Note: We also considered the possibility of a right triangle since the given lengths suggested it might be, but with the given side lengths, it is not a right triangle because the sum of the squares of the two shorter sides is not equal to the square of the longest side.