To determine if a triangle with side lengths 5 cm, 13 cm, and 12 cm is a right triangle, we can use 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.
Let's denote the side lengths as:
- [tex]\( a = 5 \)[/tex] cm
- [tex]\( b = 12 \)[/tex] cm
- [tex]\( c = 13 \)[/tex] cm
According to the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Plugging in the given values, we have:
[tex]\[ 5^2 + 12^2 = 13^2 \][/tex]
Calculate the squares:
[tex]\[ 25 + 144 = 169 \][/tex]
Now, sum the squares of the two shorter sides:
[tex]\[ 25 + 144 = 169 \][/tex]
Since [tex]\( 169 = 169 \)[/tex], we can confirm that:
[tex]\[ 5^2 + 12^2 = 13^2 \][/tex]
Therefore, the triangle with side lengths 5 cm, 13 cm, and 12 cm satisfies the condition of the Pythagorean theorem and is indeed a right triangle.
The correct explanation is:
- The triangle is a right triangle because [tex]\( 5^2 + 12^2 = 13^2 \)[/tex].
