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

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.

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.

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

1 answer

Ariel's work is assessing whether the triangle with side lengths 9, 15, and 12 forms a right triangle by using the Pythagorean theorem, which states that for 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.

Let's analyze Ariel's calculations:

  1. Identifying the sides: The given side lengths are 9, 12, and 15. The longest side is 15.
  2. Applying the Pythagorean theorem:
    • Hypotenuse (c): \( c = 15 \)
    • Other two sides (a and b): \( a = 9 \) and \( b = 12 \)

According to the Pythagorean theorem:

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

Ariel computes:

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

Now, checking her additions:

  • \( 81 + 144 = 225 \)
  • \( 225 = 225 \)

Ariel concludes \( 306 \neq 144 \), but this addition is incorrect. The calculations were not set up properly. Instead, the correct calculation should show \( 225 = 225 \).

In conclusion, Ariel's reasoning is flawed in interpreting her results. Since the correct sums actually match, we see that:

  • The correct interpretation shows that the sides can form a right triangle.

Given this analysis, the correct response should be:

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

Ariel's overall conclusion is incorrect, as she misrepresented the steps and results that led her to believe it was not a right triangle.