Question

Which of the following could NOT be the side lengths of a right triangle?

7.5, 10, and 12.5

85, 13, and 84

37, 35, and 12

24, 22, and 40

Answer 1

To determine if a set of three side lengths can be the sides of a right triangle, we can use the Pythagorean theorem. In a right triangle with sides \(a\), \(b\), and hypotenuse \(c\), the theorem states:

\[
a^2 + b^2 = c^2
\]

To apply this, we will check each set of values by identifying the longest side as the potential hypotenuse and testing the relationship.

1. **For 7.5, 10, and 12.5:**
- Longest side: 12.5
- Check: \(7.5^2 + 10^2 = 12.5^2\)
- \(56.25 + 100 = 156.25\)
- \(12.5^2 = 156.25\)
- This set satisfies the Pythagorean theorem. **It could be a right triangle.**

2. **For 85, 13, and 84:**
- Longest side: 85
- Check: \(13^2 + 84^2 = 85^2\)
- \(169 + 7056 = 7225\)
- \(85^2 = 7225\)
- This set satisfies the Pythagorean theorem. **It could be a right triangle.**

3. **For 37, 35, and 12:**
- Longest side: 37
- Check: \(35^2 + 12^2 = 37^2\)
- \(1225 + 144 = 1369\)
- \(37^2 = 1369\)
- This set satisfies the Pythagorean theorem. **It could be a right triangle.**

4. **For 24, 22, and 40:**
- Longest side: 40
- Check: \(24^2 + 22^2 = 40^2\)
- \(576 + 484 = 1600\)
- \(40^2 = 1600\)
- This set satisfies the Pythagorean theorem. **It could be a right triangle.**

Since all four sets seem to satisfy the conditions to form right triangles, I'll reassess the last option to ensure there’s no oversight, particularly looking at the relationships:

For 24, 22, and 40:

If we check again
- \(24^2 + 22^2\):
- \(576 + 484 = 1060\)
- \(40^2 = 1600\)

This does NOT satisfy the condition \(1060 \neq 1600\).

Thus, **the side lengths that could NOT form a right triangle are:**

**24, 22, and 40.**