Question

For each set of side lengths, classify the triangle based on wheather it is a right triangle.

Answer 1

For the side lengths to create a right triangle they need to satisfy Pythagorean Theorem, so $a^2 + b^2 = c^2$ where $a \ \textless \ c$ and $b \ \textless \ c$.

We can just perform this check on all the sides.

1) $7^2 + 8^2 = 49 + 64 = 113 \neq 15^2$ Not a right triangle
2) $4^2 + 10^2 = 16 + 100 = 116 \neq 11^2$ Not a right triangle
3) $8^2 + 7^2 = 64 + 49 = 113 = (\sqrt{113})^2$ Right triangle
4) $4^2 + 4^2 = 16 + 16 = 32 = (\sqrt{32})^2$ Right triangle
5) $4^2 + 4^2 = 16 + 16 = 32 \neq (\sqrt{24})^2$ Not a right triangle

There you go :P

Answer 2

Right Triangles: (Apply Pythagoras)

8,7,√113 [( 8² + 7² = (√113)² ↔ 64 + 49 = 113]

4,4,√32 [( 4² + 4² = (√32)² ↔ 16 + 16 = 32]

Not right triangles (all the remaining others):

7,8,15 [(15² ≠ 7²+8²) ↔ (225 ≠ 49+64)] so not a right triangle.
4,10,11 [(11² ≠ 4²+10²) ↔ (121 ≠ 16+100)] so not a right triangle
4,4,√24 [(√24)² ≠ 4²+4²) ↔ (24 ≠ 16+16)] so not a right triangle