To determine if the triangle with sides of lengths 5 mm, 12 mm, and 13 mm is a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
First, we identify the sides:
- The longest side is 13 mm, so we will consider it as the hypotenuse.
- The other two sides are 5 mm and 12 mm.
Now, we apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \(c\) is the hypotenuse.
Calculating: \[ 5^2 + 12^2 = 13^2 \] \[ 25 + 144 = 169 \] \[ 169 = 169 \]
Since both sides of the equation are equal, the triangle with sides of 5 mm, 12 mm, and 13 mm is indeed a right triangle.