Question

Which side lengths would form a right triangle? (1 point)
• 21, 16, & 12
• 24, 7, & 26
• 30, 24, & 18
• 9,12, & 14

Answer 1

To determine which sets of side lengths can form a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as:

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

where \( c \) is the length of the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides.

Let's evaluate each set of side lengths:

1. **21, 16, & 12**
- Hypotenuse \( c = 21 \)
- \( 21^2 = 441 \)
- \( 16^2 + 12^2 = 256 + 144 = 400 \)
- \( 441 \neq 400 \) → Not a right triangle.

2. **24, 7, & 26**
- Hypotenuse \( c = 26 \)
- \( 26^2 = 676 \)
- \( 24^2 + 7^2 = 576 + 49 = 625 \)
- \( 676 \neq 625 \) → Not a right triangle.

3. **30, 24, & 18**
- Hypotenuse \( c = 30 \)
- \( 30^2 = 900 \)
- \( 24^2 + 18^2 = 576 + 324 = 900 \)
- \( 900 = 900 \) → This is a right triangle.

4. **9, 12, & 14**
- Hypotenuse \( c = 14 \)
- \( 14^2 = 196 \)
- \( 9^2 + 12^2 = 81 + 144 = 225 \)
- \( 196 \neq 225 \) → Not a right triangle.

The only set of side lengths that can form a right triangle is:

**30, 24, & 18**.