To determine whether a triangle with side lengths 5 cm, 12 cm, and 13 cm is indeed a right triangle, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Given side lengths are:
- [tex]\( a = 5 \)[/tex] cm
- [tex]\( b = 12 \)[/tex] cm
- [tex]\( c = 13 \)[/tex] cm (hypotenuse)
We need to check if:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Substituting the given side lengths:
[tex]\[ 5^2 + 12^2 = 25 + 144 = 169 \][/tex]
Now, calculating the square of the hypotenuse:
[tex]\[ 13^2 = 169 \][/tex]
Since:
[tex]\[ 5^2 + 12^2 = 13^2 \][/tex]
Therefore, this equation holds true, confirming that the triangle is a right triangle.
Among the given options, the best explanation is:
- "The triangle is a right triangle because [tex]\( 5^2 + 12^2 = 13^2 \)[/tex]."
