To determine whether a given triangle is a right triangle, we can use the Pythagorean Theorem. The Pythagorean Theorem states that for a triangle to be a right triangle, the square of the length of the hypotenuse (the longest side of the triangle) must be equal to the sum of the squares of the lengths of the other two sides.
In this case, we are given the sides of the triangle as 12 cm, 35 cm, and 37 cm. We can assume the longest side, 37 cm, to be the hypotenuse and check if the Pythagorean relationship holds true:
1. Let's identify the sides:
- [tex]\(a = 12 \text{ cm}\)[/tex]
- [tex]\(b = 35 \text{ cm}\)[/tex]
- The hypotenuse [tex]\(c = 37 \text{ cm}\)[/tex]
2. According to the Pythagorean Theorem:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
3. Calculate the squares of the sides:
[tex]\[
a^2 = 12^2 = 144
\][/tex]
[tex]\[
b^2 = 35^2 = 1225
\][/tex]
[tex]\[
c^2 = 37^2 = 1369
\][/tex]
4. Now, sum the squares of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a^2 + b^2 = 144 + 1225 = 1369
\][/tex]
5. Compare this with the square of the hypotenuse:
[tex]\[
c^2 = 1369
\][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex] (i.e., [tex]\(1369 = 1369\)[/tex]), the triangle with sides 12 cm, 35 cm, and 37 cm satisfies the Pythagorean Theorem. Therefore, the triangle is a right triangle.
Justification: The calculations confirm that the sum of the squares of the two shorter sides is equal to the square of the longest side, which fulfills the Pythagorean Theorem. Hence, it verifies that the given triangle is indeed a right triangle.
