To determine whether a triangle with side lengths [tex]\(5 \, \text{cm}\)[/tex], [tex]\(13 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex] is a right triangle, we should use the Pythagorean theorem. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse).
Here’s the step-by-step process to check this:
1. Identify the sides:
- Let's denote the sides as: [tex]\(a = 5 \, \text{cm}\)[/tex], [tex]\(b = 12 \, \text{cm}\)[/tex], and [tex]\(c = 13 \, \text{cm}\)[/tex].
2. Apply the Pythagorean theorem:
- Hypothesis for a right triangle: [tex]\(a^2 + b^2 = c^2\)[/tex].
3. Calculate the squares of the sides:
- [tex]\(5^2 = 25\)[/tex]
- [tex]\(12^2 = 144\)[/tex]
- [tex]\(13^2 = 169\)[/tex]
4. Check the Pythagorean theorem:
- [tex]\(a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169\)[/tex]
- Compare this with [tex]\(c^2\)[/tex]:
- [tex]\(169 = 169\)[/tex]
Since the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side, the triangle with side lengths [tex]\(5 \, \text{cm}\)[/tex], [tex]\(12 \, \text{cm}\)[/tex], and [tex]\(13 \, \text{cm}\)[/tex] satisfies the Pythagorean theorem. Therefore, this triangle is a right triangle.
Thus, the best explanation for whether this triangle is a right triangle is:
- The triangle is a right triangle because [tex]\(5^2 + 12^2 = 13^2\)[/tex].
