Question

Ariel completed the work below to show that a triangle with side lengths 13, 12, and 5 do not form a right triangle.

122+132=52


144+169=25


313≠25


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



NOTE: You must answer the question in complete sentences to receive credit. You can type out your answer or upload an image of your written explanation.

(4 points)

Answer 1

Ariel's work is not correct because of a mistake in the Pythagorean theorem's application. The Pythagorean theorem states that for a right triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the length of the hypotenuse), the relationship should be \(a^2 + b^2 = c^2\).

In this case, the side lengths are 13, 12, and 5. To determine if these form a right triangle, we should identify 13 as the longest side, which we would designate as the hypotenuse (c). Thus, we need to check if:

\[
12^2 + 5^2 = 13^2.
\]

Calculating this gives:

\[
144 + 25 = 169.
\]
\[
169 = 169.
\]

Since both sides of the equation are equal, the sides 13, 12, and 5 do indeed form a right triangle. Therefore, Ariel's conclusion that these side lengths do not form a right triangle is incorrect. The proper verification of the Pythagorean theorem shows that the triangle is a right triangle.