### Video Transcript

In the following figure, find the
length of line segment π΄πΆ.

Looking at the figure, we see that
we have been given a right triangle in which the lengths of two of the sides are
known. π΄π΅ is 116 centimeters, and π΅πΆ
is 121.8 centimeters. Weβre asked to find the length of
the line segment π΄πΆ, which is the third side of this triangle.

Now, we should recall that whenever
we know the lengths of two sides in a right triangle and want to calculate the
length of the third side, we can do this by applying the Pythagorean theorem. This states that in any right
triangle, the square of the hypotenuse is equal to the sum of the squares of the two
shorter sides. If we label the two shorter sides
as having lengths π and π and the hypotenuse as having length π, then this can be
expressed as π squared plus π squared equals π squared.

In our triangle, the two shorter
sides are the sides whose lengths weβve been given and the hypotenuse is side π΄πΆ,
because itβs directly opposite the right angle. So, by the Pythagorean theorem, we
have that π΄πΆ squared is equal to 121.8 squared plus 116 squared. To solve this equation for π΄πΆ, we
first evaluate the squares and then find their sum, giving π΄πΆ squared equals
28291.24. π΄πΆ is then equal to the square
root of this value. We take only the positive value
here as π΄πΆ is a length and so must be positive. Evaluating this on a calculator
gives 168.2.

Hence, by applying the Pythagorean
theorem to calculate the length of the hypotenuse of this right triangle, weβve
found that the length of the line segment π΄πΆ is 168.2 centimeters.