Question

The side lengths of four triangles are listed below. Determine whether each triangle is a right triangle.
Select Yes or No for each set of side lengths.

6, 8, 10 A. yes B.no
3, 4, 6 A.yes B.no
10, 24, 34 A.yes B.no
8, 15, 17 A.yes B.no

Answer 1

To determine whether each set of side lengths forms a right triangle, we can use the Pythagorean theorem, which states that for a triangle with side lengths \( a \), \( b \), and hypotenuse \( c \) (where \( c \) is the longest side), the following must hold true:

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

Let's evaluate each set of side lengths:

1. **6, 8, 10**:
- Here, 10 is the longest side (\( c \)).
- Check: \( 6^2 + 8^2 = 10^2 \)
- \( 36 + 64 = 100 \)
- \( 100 = 100 \) (True)
- **Answer: A. yes**

2. **3, 4, 6**:
- Here, 6 is the longest side (\( c \)).
- Check: \( 3^2 + 4^2 = 6^2 \)
- \( 9 + 16 = 36 \)
- \( 25 \neq 36 \) (False)
- **Answer: B. no**

3. **10, 24, 34**:
- Here, 34 is the longest side (\( c \)).
- Check: \( 10^2 + 24^2 = 34^2 \)
- \( 100 + 576 = 1156 \)
- \( 676 \neq 1156 \) (False)
- **Answer: B. no**

4. **8, 15, 17**:
- Here, 17 is the longest side (\( c \)).
- Check: \( 8^2 + 15^2 = 17^2 \)
- \( 64 + 225 = 289 \)
- \( 289 = 289 \) (True)
- **Answer: A. yes**

Final answers:
1. **6, 8, 10**: A. yes
2. **3, 4, 6**: B. no
3. **10, 24, 34**: B. no
4. **8, 15, 17**: A. yes