### Video Transcript

In a triangle 𝐴𝐵𝐶, if 𝐴𝐵
squared is equal to 𝐵𝐶 squared plus 𝐴𝐶 squared, which angle is the right
angle?

In this question, we are given an
equation involving the sum of the squares of two side lengths in a triangle being
equal to the square of the length of the remaining side. And we want to determine which
angle is a right angle. We can do this by recalling the
Pythagorean theorem.

We can recall that this tells us
that in any right triangle 𝑋𝑌𝑍 with a right angle at 𝑍, we have that the square
of the length of the hypotenuse is equal to the sum of the squares of the lengths of
the two shorter sides. So 𝑋𝑌 squared is equal to 𝑌𝑍
squared plus 𝑋𝑍 squared. An interesting and useful fact
about the Pythagorean theorem is that it works in both directions. If we have three side lengths in a
triangle that satisfy the equation 𝑋𝑌 squared is equal to 𝑌𝑍 squared plus 𝑋𝑍
squared, then it must be a right triangle with a right angle opposite the
hypotenuse.

We are told that 𝐴𝐵 squared is
equal to 𝐵𝐶 squared plus 𝐴𝐶 squared. So we must have that these points
form a right triangle with right angle opposite the hypotenuse. Since 𝐴𝐵 is the hypotenuse, the
right angle must be opposite this side at vertex 𝐶. Hence, the right angle in the
triangle is angle 𝐶.