To determine if a triangle with side lengths of [tex]\( 5 \, \text{cm} \)[/tex], [tex]\( 13 \, \text{cm} \)[/tex], and [tex]\( 12 \, \text{cm} \)[/tex] is a right triangle, we can use the Pythagorean theorem. The 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.
Let's identify the sides:
- Side [tex]\( a = 5 \, \text{cm} \)[/tex]
- Side [tex]\( b = 12 \, \text{cm} \)[/tex]
- Hypotenuse [tex]\( c = 13 \, \text{cm} \)[/tex] (since it is the longest side)
According to the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's check:
[tex]\[ 5^2 + 12^2 = 13^2 \][/tex]
Calculate each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Now, sum the squares of the legs:
[tex]\[ 25 + 144 = 169 \][/tex]
Compare it with the square of the hypotenuse:
[tex]\[ 169 = 169 \][/tex]
Since both sides of the equation are equal, this confirms that the triangle with side lengths [tex]\( 5 \, \text{cm} \)[/tex], [tex]\( 13 \, \text{cm} \)[/tex], and [tex]\( 12 \, \text{cm} \)[/tex] satisfies the Pythagorean theorem and therefore is a right triangle.
Thus, the correct explanation is:
"The triangle is a right triangle because [tex]\( 5^2 + 12^2 = 13^2 \)[/tex]."
