To determine whether each set of numbers is a Pythagorean triple, we need to check if the numbers satisfy the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with sides \(a\), \(b\), and hypotenuse \(c\):
\[
a^2 + b^2 = c^2
\]
Here, \(c\) is the largest number in the set. We will test each set of numbers to see if they satisfy this equation.
---
1) 6, 8, 10
- Identify the largest number: \(c = 10\)
- Check if \(6^2 + 8^2 = 10^2\):
\[
6^2 = 36, \quad 8^2 = 64, \quad 10^2 = 100
\]
\[
6^2 + 8^2 = 36 + 64 = 100
\]
\[
10^2 = 100
\]
Since \(6^2 + 8^2 = 10^2\), this is a Pythagorean triple.
Answer: Yes
---
2) 16, 7, 3
- Identify the largest number: \(c = 16\)
- Check if \(7^2 + 3^2 = 16^2\):
\[
7^2 = 49, \quad 3^2 = 9, \quad 16^2 = 256
\]
\[
7^2 + 3^2 = 49 + 9 = 58
\]
\[
16^2 = 256
\]
Since \(7^2 + 3^2 \neq 16^2\), this is not a Pythagorean triple.
Answer: No
---
3) 32, 21, 26
- Identify the largest number: \(c = 32\)
- Check if \(21^2 + 26^2 = 32^2\):
\[
21^2 = 441, \quad 26^2 = 676, \quad 32^2 = 1024
\]
\[
21^2 + 26^2 = 441 + 676 = 1117
\]
\[
32^2 = 1024
\]
Since \(21^2 + 26^2 \neq 32^2\), this is not a Pythagorean triple.
Answer: No
---
4) 20, 25, 15
- Identify the largest number: \(c = 25\)
- Check if \(20^2 + 15^2 = 25^2\):
\[
20^2 = 400, \quad 15^2 = 225, \quad 25^2 = 625
\]
\[
20^2 + 15^2 = 400 + 225 = 625
\]
\[
25^2 = 625
\]
Since \(20^2 + 15^2 = 25^2\), this is a Pythagorean triple.
Answer: Yes
---
5) 30, 24, 18
- Identify the largest number: \(c = 30\)
- Check if \(24^2 + 18^2 = 30^2\):
\[
24^2 = 576, \quad 18^2 = 324, \quad 30^2 = 900
\]
\[
24^2 + 18^2 = 576 + 324 = 900
\]
\[
30^2 = 900
\]
Since \(24^2 + 18^2 = 30^2\), this is a Pythagorean triple.
Answer: Yes
---
6) 2, 6, 9
- Identify the largest number: \(c = 9\)
- Check if \(2^2 + 6^2 = 9^2\):
\[
2^2 = 4, \quad 6^2 = 36, \quad 9^2 = 81
\]
\[
2^2 + 6^2 = 4 + 36 = 40
\]
\[
9^2 = 81
\]
Since \(2^2 + 6^2 \neq 9^2\), this is not a Pythagorean triple.
Answer: No
---
7) 4, 11, 5
- Identify the largest number: \(c = 11\)
- Check if \(4^2 + 5^2 = 11^2\):
\[
4^2 = 16, \quad 5^2 = 25, \quad 11^2 = 121
\]
\[
4^2 + 5^2 = 16 + 25 = 41
\]
\[
11^2 = 121
\]
Since \(4^2 + 5^2 \neq 11^2\), this is not a Pythagorean triple.
Answer: No
---
8) 28, 35, 21
- Identify the largest number: \(c = 35\)
- Check if \(28^2 + 21^2 = 35^2\):
\[
28^2 = 784, \quad 21^2 = 441, \quad 35^2 = 1225
\]
\[
28^2 + 21^2 = 784 + 441 = 1225
\]
\[
35^2 = 1225
\]
Since \(28^2 + 21^2 = 35^2\), this is a Pythagorean triple.
Answer: Yes
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
1) & \text{Yes} \\
2) & \text{No} \\
3) & \text{No} \\
4) & \text{Yes} \\
5) & \text{Yes} \\
6) & \text{No} \\
7) & \text{No} \\
8) & \text{Yes} \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem number sets.
