Question

Which side lengths would form a right triangle?

A. 21, 16, & 12
B. 9, 12, & 14
C. 24, 7, & 26
D. 30, 24, & 18

Answer 1

To determine which set 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. Specifically:

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

Where \( c \) is the length of the hypotenuse.

Let's examine each option:

**A. 21, 16, & 12**
- Sort the sides: \( 21 \) (hypotenuse), \( 16 \), and \( 12 \)
- Check: \( 21^2 = 16^2 + 12^2 \)
- \( 441 \) vs \( 256 + 144 \)
- \( 441 = 400 \) (not true)

**B. 9, 12, & 14**
- Sort the sides: \( 14 \) (hypotenuse), \( 12 \), and \( 9 \)
- Check: \( 14^2 = 12^2 + 9^2 \)
- \( 196 \) vs \( 144 + 81 \)
- \( 196 = 225 \) (not true)

**C. 24, 7, & 26**
- Sort the sides: \( 26 \) (hypotenuse), \( 24 \), and \( 7 \)
- Check: \( 26^2 = 24^2 + 7^2 \)
- \( 676 \) vs \( 576 + 49 \)
- \( 676 = 625 \) (not true)

**D. 30, 24, & 18**
- Sort the sides: \( 30 \) (hypotenuse), \( 24 \), and \( 18 \)
- Check: \( 30^2 = 24^2 + 18^2 \)
- \( 900 \) vs \( 576 + 324 \)
- \( 900 = 900 \) (true)

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

**D. 30, 24, & 18**