Question

hich of the following sets of numbers could represent the three sides of a right triangle?
Answer
Multiple Choice Answers
left curly bracket, 40, comma, 41, comma, 58, right curly bracket{40,41,58}
left curly bracket, 16, comma, 30, comma, 34, right curly bracket{16,30,34}
left curly bracket, 14, comma, 84, comma, 85, right curly bracket{14,84,85}
left curly bracket, 39, comma, 52, comma, 66, right curly bracket{39,52,66}

Answer 1

To determine which set of numbers could represent the three sides of 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. In formula form, it is:

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

where \(c\) is the longest side.

Let's evaluate each set of numbers:

1. **Set \( \{40, 41, 58\} \)**:
- The longest side is 58.
- Check: \( 58^2 \stackrel{?}{=} 40^2 + 41^2 \)
- \( 58^2 = 3364 \)
- \( 40^2 + 41^2 = 1600 + 1681 = 3281 \)
- Not a right triangle.

2. **Set \( \{16, 30, 34\} \)**:
- The longest side is 34.
- Check: \( 34^2 \stackrel{?}{=} 16^2 + 30^2 \)
- \( 34^2 = 1156 \)
- \( 16^2 + 30^2 = 256 + 900 = 1156 \)
- This is a right triangle.

3. **Set \( \{14, 84, 85\} \)**:
- The longest side is 85.
- Check: \( 85^2 \stackrel{?}{=} 14^2 + 84^2 \)
- \( 85^2 = 7225 \)
- \( 14^2 + 84^2 = 196 + 7056 = 7252 \)
- Not a right triangle.

4. **Set \( \{39, 52, 66\} \)**:
- The longest side is 66.
- Check: \( 66^2 \stackrel{?}{=} 39^2 + 52^2 \)
- \( 66^2 = 4356 \)
- \( 39^2 + 52^2 = 1521 + 2704 = 4225 \)
- Not a right triangle.

From the above calculations, the only set that represents the sides of a right triangle is:

\[
\{16, 30, 34\}
\]