Question Video: Finding the Altitude of a Right Triangle within a Trapezoid of Known Area | Nagwa Question Video: Finding the Altitude of a Right Triangle within a Trapezoid of Known Area | Nagwa

# Question Video: Finding the Altitude of a Right Triangle within a Trapezoid of Known Area Mathematics

Given that the area of the trapezoid π΄π΅πΆπ· is 9,522 cmΒ², determine the length of the side π΅πΉ.

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### Video Transcript

Given that the area of the trapezoid π΄π΅πΆπ· is 9,522 square centimeters, determine the length of the side π΅πΉ.

In this example, we want to find π΅πΉ, which is the perpendicular projection to πΉ, from the right angle at π΅ in triangle π΄π΅πΈ. And to find this length, weβre going to use the right triangle altitude theorem to find the lengths of sides π΄πΉ and πΈπΉ and then its corollary to find the square of the side we want, which is side π΅πΉ. And taking the square root will give us the length of π΅πΉ.

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

Now, weβre told in the question that the area of the trapezoid π΄π΅πΆπ· 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 πΆπΈ plus πΈπ΅. Thatβs 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 thatβs 69 centimeters long.

We know the area, 9,522, and the only thing we donβt know in this formula for the area of trapezoid π΄π΅πΆπ· 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΅πΈ. Thatβs π΄π΅ equals 92 centimeters and πΈπ΅ equals 69 centimeters. And since this is a right triangle, we can use the Pythagorean theorem to find the third side π΄πΈ. We have π΄π΅ squared plus πΈπ΅ squared equals π΄πΈ squared. Thatβs 92 squared plus 69 squared equals π΄πΈ squared.

Evaluating the left-hand side gives 13,225 equals π΄πΈ squared. And taking the positive square root on both sides, positive since lengths are positive, we have π΄πΈ 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, itβs side π΅πΉ that we want to find. And to use the corollary to find this, we need lengths π΄πΉ and πΈπΉ. But since we now have side lengths π΄π΅, πΈπ΅, and π΄πΈ, we can find these easily using the two formulae in the theorem.

In the first case, we have 92 squared, thatβs π΄π΅ squared, equals π΄πΉ times 115, which is π΄πΈ. And dividing both sides by 115, then evaluating the left-hand side, we have π΄πΉ equals 73.6. Now, for the second equation in the altitude theorem, we have 69 squared, thatβs πΈπ΅ squared, equals πΈπΉ times 115, which is πΈπ΄. Then, dividing both sides by 115 and evaluating, we have πΈπΉ 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΅πΉ. This gives π΅πΉ 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΅πΉ equal to 55.2. Hence, using the right triangle altitude theorem and its corollary, we find the side length π΅πΉ equals 55.2 centimeters.