Ariel completed the work below to show that a triangle with side lengths 9, 15, and 12 do not form a right triangle.

Question

92+152=122


81+225=144


306≠144


Is Ariel's work correct? Why or why not?



NOTE: On the sample work, this is a multiple choice question, but on the test, you will be asked to explain your answer in complete sentences. Think about WHY she is right or wrong when completing this question, so you are prepared for the test.

(1 point)
Responses

No, when setting up the Pythagorean Theorem, the longest side needs to be by itself.
No, when setting up the Pythagorean Theorem, the longest side needs to be by itself.

No, she squared the numbers instead of multiplying each of them by 2.
No, she squared the numbers instead of multiplying each of them by 2.

Yes, Ariel's work is correct and this is a right triangle.
Yes, Ariel's work is correct and this is a right triangle.

Yes, Ariel's work is correct and this is not a right triangle.

Answer 1

Ariel's work is correct, and this is not a right triangle.

To determine if a triangle is a right triangle, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the sides of the triangle are 9, 12, and 15. The longest side is 15. According to the Pythagorean Theorem, we set it up as follows:

\( 9^2 + 12^2 = 15^2 \)

Calculating the squares:

• \( 9^2 = 81 \)
• \( 12^2 = 144 \)
• \( 15^2 = 225 \)

Now, we add the squares of the two shorter sides:

\( 81 + 144 = 225 \)

Since \( 225 = 225 \), this indicates that the triangle with side lengths 9, 12, and 15 is a right triangle.

However, Ariel incorrectly concluded that the triangle is not a right triangle because she miscalculated and compared the values incorrectly (306 ≠ 144 is not the correct comparison). Therefore, her conclusion is incorrect, although her process of attempting to use the Pythagorean Theorem was on the right track. Thus, the correct response is "No, when setting up the Pythagorean Theorem, the longest side needs to be by itself," reflecting that she incorrectly set up her equation based on the values she summed.