To determine which triangle is a right triangle among the given options, we need to apply the Pythagorean theorem. The Pythagorean 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. Mathematically, if [tex]\(c\)[/tex] is the hypotenuse and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the other two sides, then:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's examine each set of side lengths:
Option A: 6, 8, 12
- First, we identify the largest side, which is 12 (potential hypotenuse).
- Calculate [tex]\(6^2 + 8^2\)[/tex]:
[tex]\[ 6^2 + 8^2 = 36 + 64 = 100 \][/tex]
- Compare with [tex]\(12^2\)[/tex]:
[tex]\[ 12^2 = 144 \][/tex]
- Since [tex]\(100 \neq 144\)[/tex], this is not a right triangle.
Option B: 13, 14, 15
- The largest side is 15 (potential hypotenuse).
- Calculate [tex]\(13^2 + 14^2\)[/tex]:
[tex]\[ 13^2 = 169 \][/tex]
[tex]\[ 14^2 = 196 \][/tex]
[tex]\[ 169 + 196 = 365 \][/tex]
- Compare with [tex]\(15^2\)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
- Since [tex]\(365 \neq 225\)[/tex], this is not a right triangle.
Option C: 15, 20, 25
- The largest side is 25 (potential hypotenuse).
- Calculate [tex]\(15^2 + 20^2\)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ 225 + 400 = 625 \][/tex]
- Compare with [tex]\(25^2\)[/tex]:
[tex]\[ 25^2 = 625 \][/tex]
- Since [tex]\(625 = 625\)[/tex], this is a right triangle.
Option D: 12, 14, 16
- The largest side is 16 (potential hypotenuse).
- Calculate [tex]\(12^2 + 14^2\)[/tex]:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 14^2 = 196 \][/tex]
[tex]\[ 144 + 196 = 340 \][/tex]
- Compare with [tex]\(16^2\)[/tex]:
[tex]\[ 16^2 = 256 \][/tex]
- Since [tex]\(340 \neq 256\)[/tex], this is not a right triangle.
Based on these calculations, only Option C (15, 20, 25) forms a right triangle.
Therefore, the right triangle is:
C. 15-20-25
