To determine if a triangle with side lengths [tex]\(5\)[/tex] cm, [tex]\(12\)[/tex] cm, and [tex]\(13\)[/tex] cm is a right triangle, we need to verify if it satisfies the Pythagorean Theorem. The Pythagorean Theorem states that for a right triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the following equation holds true:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Given the side lengths [tex]\(5\)[/tex] cm, [tex]\(12\)[/tex] cm, and [tex]\(13\)[/tex] cm, we can assume that [tex]\(5\)[/tex] cm and [tex]\(12\)[/tex] cm are the legs ([tex]\(a\)[/tex] and [tex]\(b\)[/tex]), and [tex]\(13\)[/tex] cm is the hypotenuse ([tex]\(c\)[/tex]). We need to check the following:
[tex]\[ 5^2 + 12^2 = 13^2 \][/tex]
First, calculate [tex]\(5^2\)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
Next, calculate [tex]\(12^2\)[/tex]:
[tex]\[ 12^2 = 144 \][/tex]
Now, sum these squares:
[tex]\[ 25 + 144 = 169 \][/tex]
Then, calculate [tex]\(13^2\)[/tex]:
[tex]\[ 13^2 = 169 \][/tex]
We see that:
[tex]\[ 5^2 + 12^2 = 13^2 = 169 \][/tex]
Since [tex]\(5^2 + 12^2 = 13^2\)[/tex], the given side lengths satisfy the Pythagorean Theorem. Therefore, the triangle is a right triangle.
So, the correct explanation is:
The triangle is a right triangle because [tex]\(5^2 + 12^2 = 13^2\)[/tex].
