Which of the following does NOT represent the lengths of the sides of a right triangle?

Select THREE correct answers.
A.9
, 12
, and 18

B.6
, 10
, and 14

C.10
, 24
, and 26

D.15
, 20
, and 25

E.18
, 20
, and 28

To determine which sets of numbers do **not** represent the lengths of the sides of a right triangle, we use the **Pythagorean theorem**, which states that for a right triangle with sides \(a\), \(b\), and hypotenuse \(c\), the following must be true:

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

We will apply this to each set of numbers:

### A. \(9, 12, 18\)
Check if \(9^2 + 12^2 = 18^2\):
\[
9^2 + 12^2 = 81 + 144 = 225
\]
\[
18^2 = 324
\]
Since \(225 \neq 324\), this **does not** form a right triangle.

### B. \(6, 10, 14\)
Check if \(6^2 + 10^2 = 14^2\):
\[
6^2 + 10^2 = 36 + 100 = 136
\]
\[
14^2 = 196
\]
Since \(136 \neq 196\), this **does not** form a right triangle.

### C. \(10, 24, 26\)
Check if \(10^2 + 24^2 = 26^2\):
\[
10^2 + 24^2 = 100 + 576 = 676
\]
\[
26^2 = 676
\]
Since \(676 = 676\), this **does** form a right triangle.

### D. \(15, 20, 25\)
Check if \(15^2 + 20^2 = 25^2\):
\[
15^2 + 20^2 = 225 + 400 = 625
\]
\[
25^2 = 625
\]
Since \(625 = 625\), this **does** form a right triangle.

### E. \(18, 20, 28\)
Check if \(18^2 + 20^2 = 28^2\):
\[
18^2 + 20^2 = 324 + 400 = 724
\]
\[
28^2 = 784
\]
Since \(724 \neq 784\), this **does not** form a right triangle.

### Conclusion:
The sets of numbers that do **not** represent the lengths of the sides of a right triangle are:
- A. \(9, 12, 18\)
- B. \(6, 10, 14\)
- E. \(18, 20, 28\)

These are the correct answers.

Would you like more details on any part of this solution? 

Here are 5 related questions for further practice:
1. What is the Pythagorean theorem, and why is it used for right triangles?
2. Can any three positive integers form the sides of a right triangle?
3. How do you find the hypotenuse if the other two sides of a right triangle are known?
4. What are Pythagorean triples, and can they help in solving these problems?
5. How can you determine if a triangle is obtuse or acute using the Pythagorean theorem?

**Tip:** A quick way to check if three numbers can form a right triangle is to see if they fit common Pythagorean triples like (3, 4, 5) or their multiples (e.g., 6, 8, 10).

Which of the following does NOT represent the lengths of the sides of a right triangle? Select THREE correct answers. A. 9, 12, and 18 B. 6, 10, and 14 C. 10, 24, and 26 D. 15, 20, and 25 E. 18, 20, and 28

Learn how to identify sets of numbers that do not form the sides of a right triangle using the Pythagorean theorem. This problem covers core geometry concepts and is suitable for students in Grades 8-10.

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