To determine whether a triangle with side lengths [tex]\(5 \, \text{cm}, 13 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex] is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the following relationship holds:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
1. Let's denote the sides as:
- [tex]\( a = 5 \, \text{cm} \)[/tex]
- [tex]\( b = 12 \, \text{cm} \)[/tex]
- [tex]\( c = 13 \, \text{cm} \)[/tex]
2. Calculate the squares of each side:
- [tex]\( a^2 = 5^2 = 25 \)[/tex]
- [tex]\( b^2 = 12^2 = 144 \)[/tex]
- [tex]\( c^2 = 13^2 = 169 \)[/tex]
3. Now check whether [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ 25 + 144 = 169 \][/tex]
Indeed, [tex]\( 25 + 144 = 169 \)[/tex], which matches [tex]\( c^2 \)[/tex].
Since [tex]\( a^2 + b^2 = c^2 \)[/tex] is satisfied, this confirms that the triangle with side lengths [tex]\(5 \, \text{cm}\)[/tex], [tex]\(12 \, \text{cm}\)[/tex], and [tex]\(13 \, \text{cm}\)[/tex] is a right triangle.
Thus, the correct explanation is:
"The triangle is a right triangle because [tex]\(5^2 + 12^2 = 13^2\)[/tex]."
