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).