Question

Which sets of side lengths represent Pythagorean triples? Check all that apply.
1, 2, 5
6, 8, 14
8, 15, 17
10, 24, 26
15, 20, 30
28, 45, 53

Answer 1

To determine which sets of side lengths represent Pythagorean triples, we need to check if the sets satisfy the Pythagorean theorem, which states that for any right triangle with sides \( a \) and \( b \), and hypotenuse \( c \):

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

We will evaluate each set of side lengths:

1. **1, 2, 5**:
\[
1^2 + 2^2 = 1 + 4 = 5 \quad \text{and} \quad 5^2 = 25 \quad \text{(not a triple)}
\]

2. **6, 8, 14**:
\[
6^2 + 8^2 = 36 + 64 = 100 \quad \text{and} \quad 14^2 = 196 \quad \text{(not a triple)}
\]

3. **8, 15, 17**:
\[
8^2 + 15^2 = 64 + 225 = 289 \quad \text{and} \quad 17^2 = 289 \quad \text{(this is a triple)}
\]

4. **10, 24, 26**:
\[
10^2 + 24^2 = 100 + 576 = 676 \quad \text{and} \quad 26^2 = 676 \quad \text{(this is a triple)}
\]

5. **15, 20, 30**:
\[
15^2 + 20^2 = 225 + 400 = 625 \quad \text{and} \quad 30^2 = 900 \quad \text{(not a triple)}
\]

6. **28, 45, 53**:
\[
28^2 + 45^2 = 784 + 2025 = 2809 \quad \text{and} \quad 53^2 = 2809 \quad \text{(this is a triple)}
\]

In summary, the sets of side lengths that represent Pythagorean triples are:

- **8, 15, 17**
- **10, 24, 26**
- **28, 45, 53**