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, 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.

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.

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

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

Answer 1

Ariel's work is correct in demonstrating that the triangle with side lengths 9, 15, and 12 does not form a right triangle. In a right triangle, the Pythagorean Theorem states that the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides.

In this case:

1. The longest side is 15.
2. The squares of the sides are calculated as follows:
   • \(9^2 = 81\)
   • \(12^2 = 144\)
   • \(15^2 = 225\)

Next, we check the Pythagorean Theorem:

\[ 9^2 + 12^2 = 81 + 144 = 225 \] This means that: \[ 9^2 + 12^2 \text{ indeed equals } 15^2. \]

Since \(81 + 144 = 225\), we see that Ariel's concern regarding the triangle not being a right triangle was incorrect based on her calculations. In fact, it does satisfy the Pythagorean theorem for \(9\), \(12\), and \(15\), meaning it is a right triangle.

Given this, the correct response is:

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