### Video Transcript

Given that the area of the trapezoid π΄π΅πΆπ· is 9,522 square centimeters, determine the length of the side π΅πΉ.

In this example, we want to find π΅πΉ, which is the perpendicular projection to πΉ, from the right angle at π΅ in triangle π΄π΅πΈ. And to find this length, weβre going to use the right triangle altitude theorem to find the lengths of sides π΄πΉ and πΈπΉ and then its corollary to find the square of the side we want, which is side π΅πΉ. And taking the square root will give us the length of π΅πΉ.

To do all this, weβre going to need to find each of the side lengths of triangle π΄π΅πΈ. And for this, we first need to extract some information from the diagram. So we can see that sides π·π΄ and πΆπΈ are parallel, as are sides πΆπ· and πΈπ΄. We can therefore conclude that π΄π·πΆπΈ is a parallelogram within the trapezoid π΄π΅πΆπ·. This means that sides π·π΄ and πΆπΈ must have equal length. And since weβre given that π·π΄ is 69 centimeters, then πΆπΈ must also have length 69 centimeters. We can also see that side πΈπ΅ of triangle π΄π΅πΈ has the same length as side πΆπΈ. So this too must have length 69 centimeters.

Now, weβre told in the question that the area of the trapezoid π΄π΅πΆπ· is 9522 square centimeters. And we know that the area of a trapezoid with vertical height β, base π, and parallel side π is given by π plus π over two multiplied by β. Comparing to the given trapezoid, we can see that the base is equal to πΆπΈ plus πΈπ΅. Thatβs 69 plus 69, which is 138. We see also that the side parallel to this is side π·π΄. So we have the parallel π is our side π·π΄, and thatβs 69 centimeters long.

We know the area, 9,522, and the only thing we donβt know in this formula for the area of trapezoid π΄π΅πΆπ· is the perpendicular height, β. So, now to solve for β, we can multiply both sides by two and evaluate inside the parentheses, giving 19,044 equals 207β. Finally, dividing both sides by 207, we find β equals 92 centimeters.

Now, in the given trapezoid, the height β is actually the side length π΄π΅. So we have side length π΄π΅ equal to 92 centimeters. So, now clearing some space, we see that we have the lengths of two of the sides of triangle π΄π΅πΈ. Thatβs π΄π΅ equals 92 centimeters and πΈπ΅ equals 69 centimeters. And since this is a right triangle, we can use the Pythagorean theorem to find the third side π΄πΈ. We have π΄π΅ squared plus πΈπ΅ squared equals π΄πΈ squared. Thatβs 92 squared plus 69 squared equals π΄πΈ squared.

Evaluating the left-hand side gives 13,225 equals π΄πΈ squared. And taking the positive square root on both sides, positive since lengths are positive, we have π΄πΈ equal to 115 centimeters.

Marking this on our diagram, we can now turn our attention to the right triangle altitude theorem and its corollary. Remember, itβs side π΅πΉ that we want to find. And to use the corollary to find this, we need lengths π΄πΉ and πΈπΉ. But since we now have side lengths π΄π΅, πΈπ΅, and π΄πΈ, we can find these easily using the two formulae in the theorem.

In the first case, we have 92 squared, thatβs π΄π΅ squared, equals π΄πΉ times 115, which is π΄πΈ. And dividing both sides by 115, then evaluating the left-hand side, we have π΄πΉ equals 73.6. Now, for the second equation in the altitude theorem, we have 69 squared, thatβs πΈπ΅ squared, equals πΈπΉ times 115, which is πΈπ΄. Then, dividing both sides by 115 and evaluating, we have πΈπΉ equal to 41.4.

We can check our side lengths are correct by summing our two answers. The sum does equal 115, so we can assume our side lengths are correct. Now we can use these lengths in the corollary to find the side length π΅πΉ. This gives π΅πΉ squared equal to 73.6 times 41.4, which is 3,047.04. So, now taking the positive square root on both sides, we have π΅πΉ equal to 55.2. Hence, using the right triangle altitude theorem and its corollary, we find the side length π΅πΉ equals 55.2 centimeters.