First, let's understand what it means for a triangle to be a right triangle. A triangle is a right triangle if it satisfies the Pythagorean theorem, which states [tex]\( a^2 + b^2 = c^2 \)[/tex], where [tex]\( c \)[/tex] is the hypotenuse (the longest side of the triangle) and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the other two sides.
Given the side lengths: [tex]\( 5 \, \text{cm} \)[/tex], [tex]\( 13 \, \text{cm} \)[/tex], and [tex]\( 12 \, \text{cm} \)[/tex]:
1. Identify the longest side, which will be the hypotenuse ([tex]\( c \)[/tex]). In this case, it is [tex]\( 13 \, \text{cm} \)[/tex].
2. Use the Pythagorean theorem to check if the given sides satisfy the condition:
[tex]\[
a^2 + b^2 \stackrel{?}{=} c^2
\][/tex]
Here, [tex]\( a = 5 \, \text{cm} \)[/tex], [tex]\( b = 12 \, \text{cm} \)[/tex], and [tex]\( c = 13 \, \text{cm} \)[/tex].
3. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
4. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[
12^2 = 144
\][/tex]
5. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[
13^2 = 169
\][/tex]
6. Add [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] to verify if the left-hand side of the equation equals the right-hand side:
[tex]\[
25 + 144 = 169
\][/tex]
7. Since the left-hand side [tex]\( 25 + 144 \)[/tex] equals the right-hand side, [tex]\( 169 \)[/tex], we can conclude the triangle satisfies the Pythagorean theorem. Therefore, the side lengths form a right triangle.
So, the best explanation is:
The triangle is a right triangle because [tex]\( 5^2 + 12^2 = 13^2 \)[/tex].
