10 The following shows the use of the converse of the Pythagorean Theorem using side lengths of four different triangles. Based on the converse, which of the following sets of side lengths is a right triangle?
2^{2}+3^{2}=4^{2}
5^{2}+12^{2}=13^{7}
6^{2}+9^{2}=10^{2}
10^{2}+11^{2}=15^{2}
The converse of the Pythagorean Theorem states that if the sum of the squares of two shorter sides of a triangle is equal to the square of the longest side, then the triangle is a right triangle.
Using this information, in the given sets of side lengths, the only one that forms a right triangle is:
6^{2}+9^{2}=10^{2}
6^{2}+9^{2}=36+81=117
10^{2}=100
Since 6^{2}+9^{2}=117 and 10^{2}=100, this set of side lengths forms a right triangle.