Question

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

12^(2) + 13^(2) = 5^(2)
144+169=25
313 (crossed out)= 25

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

Answer 1

Ariel's work is not correct, but the conclusion that the triangle with side lengths 13, 12, and 5 does not form a right triangle is correct.

To determine if three sides can form a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the following relationship holds:

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

In the case of the triangle with side lengths 13, 12, and 5, we should identify the largest side to serve as the potential hypotenuse. Here, 13 is the largest side, and thus we can denote:

- \( a = 5 \)
- \( b = 12 \)
- \( c = 13 \)

Now we apply the Pythagorean theorem:

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

Calculating each side:

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

This shows that when using 5 and 12 as the legs and 13 as the hypotenuse, the equation holds true. Therefore, these sides do form a right triangle.

Ariel incorrectly computed \( 12^2 + 13^2 = 5^2 \), which is not a valid application of the theorem. Instead, it should have been \( 5^2 + 12^2 = 13^2 \). Thus, while Ariel concluded that these side lengths do not form a right triangle, the correct conclusion is that they do form a right triangle.