To show that the side lengths of 12, 16, and 20 form a right triangle, we need to verify that the Pythagorean theorem holds for these side lengths. The Pythagorean theorem states that for any right triangle, the sum of the squares of the lengths of the legs (the shorter sides) equals the square of the length of the hypotenuse (the longest side).
Let's label the sides of the triangle as follows:
a = 12 (one of the legs)
b = 16 (the other leg)
c = 20 (the hypotenuse)
Then we can plug these values into the Pythagorean theorem:
a^2 + b^2 = c^2
(12)^2 + (16)^2 = (20)^2
144 + 256 = 400
400 = 400
Since both sides are equal, this verifies that the side lengths of 12, 16, and 20 form a right triangle.