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 sets of numbers represent the sides of a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with sides \( a \), \( b \), and hypotenuse \( c \), the equation \( a^2 + b^2 = c^2 \) must hold true.

Let's evaluate each set of numbers:

1. **{40, 41, 58}**
- Assume \( c = 58 \) (the largest number).
- Check if \( 40^2 + 41^2 = 58^2 \):
- \( 40^2 = 1600 \)
- \( 41^2 = 1681 \)
- \( 58^2 = 3364 \)
- \( 1600 + 1681 = 3281 \), which does not equal \( 3364 \).
- **Not a right triangle.**

2. **{16, 30, 34}**
- Assume \( c = 34 \).
- Check if \( 16^2 + 30^2 = 34^2 \):
- \( 16^2 = 256 \)
- \( 30^2 = 900 \)
- \( 34^2 = 1156 \)
- \( 256 + 900 = 1156 \), which equals \( 1156 \).
- **This set is a right triangle.**

3. **{14, 84, 85}**
- Assume \( c = 85 \).
- Check if \( 14^2 + 84^2 = 85^2 \):
- \( 14^2 = 196 \)
- \( 84^2 = 7056 \)
- \( 85^2 = 7225 \)
- \( 196 + 7056 = 7252 \), which does not equal \( 7225 \).
- **Not a right triangle.**

4. **{39, 52, 66}**
- Assume \( c = 66 \).
- Check if \( 39^2 + 52^2 = 66^2 \):
- \( 39^2 = 1521 \)
- \( 52^2 = 2704 \)
- \( 66^2 = 4356 \)
- \( 1521 + 2704 = 4225 \), which does not equal \( 4356 \).
- **Not a right triangle.**

From the analysis above, the only set that could represent the sides of a right triangle is **{16, 30, 34}**.