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?

Answer 1

Ariel's conclusion that the triangle with sides 13, 12, and 5 does not form a right triangle is indeed correct, but the way the calculations were presented does not make sense based on the Pythagorean theorem.

To determine if the triangle is a right triangle, we should use the Pythagorean theorem, which states that for a triangle with side lengths \( a \), \( b \), and \( c \) (where \( c \) is the longest side), if it is a right triangle, then:

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

In this case, let's denote the sides as follows:
- \( a = 5 \)
- \( b = 12 \)
- \( c = 13 \)

We need to check if:

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

Calculating the squares, we have:

\[ 5^2 = 25 \]
\[ 12^2 = 144 \]
\[ 13^2 = 169 \]

Now, adding the squares of the shorter sides:

\[ 5^2 + 12^2 = 25 + 144 = 169 \]

And comparing it to the square of the longest side:

\[ 13^2 = 169 \]

Since:

\[ 25 + 144 = 169 \]

This shows that:

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

Thus, the triangle with sides 5, 12, and 13 does indeed **form a right triangle**, as the equation is balanced. Therefore, while Ariel's conclusion is that the triangle does not form a right triangle, the calculations shown seem to be incorrect and misleading.

In summary, the triangle with sides 5, 12, and 13 is a right triangle.