To determine whether a triangle with side lengths of [tex]\(5 \, \text{cm}\)[/tex], [tex]\(12 \, \text{cm}\)[/tex], and [tex]\(13 \, \text{cm}\)[/tex] is a right triangle, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse ([tex]\(c\)[/tex]) is equal to the sum of the squares of the other two sides ([tex]\(a\)[/tex] and [tex]\(b\)[/tex]):
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Here, it seems that the side lengths [tex]\(5 \, \text{cm}\)[/tex] and [tex]\(12 \, \text{cm}\)[/tex] are the legs of the triangle, and [tex]\(13 \, \text{cm}\)[/tex] is the hypotenuse. We will check if this satisfies the Pythagorean theorem:
1. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
2. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ 12^2 = 144 \][/tex]
3. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[ 13^2 = 169 \][/tex]
4. Now, sum the squares of the legs:
[tex]\[ 5^2 + 12^2 = 25 + 144 = 169 \][/tex]
5. Compare this to the square of the hypotenuse:
[tex]\[ 169 = 169 \][/tex]
The equality holds, so this shows that [tex]\(5^2 + 12^2 = 13^2\)[/tex].
Therefore, the correct explanation is:
"The triangle is a right triangle because [tex]\(5^2 + 12^2 = 13^2\)[/tex]."
