Given that the area of the trapezoid 𝐴𝐵𝐶𝐷 is 9,522 square centimeters, determine the length of the side 𝐵𝐹.
In this example, we want to find 𝐵𝐹, which is the perpendicular projection to 𝐹, from the right angle at 𝐵 in triangle 𝐴𝐵𝐸. And to find this length, we’re going to use the right triangle altitude theorem to find the lengths of sides 𝐴𝐹 and 𝐸𝐹 and then its corollary to find the square of the side we want, which is side 𝐵𝐹. And taking the square root will give us the length of 𝐵𝐹.
To do all this, we’re going to need to find each of the side lengths of triangle 𝐴𝐵𝐸. And for this, we first need to extract some information from the diagram. So we can see that sides 𝐷𝐴 and 𝐶𝐸 are parallel, as are sides 𝐶𝐷 and 𝐸𝐴. We can therefore conclude that 𝐴𝐷𝐶𝐸 is a parallelogram within the trapezoid 𝐴𝐵𝐶𝐷. This means that sides 𝐷𝐴 and 𝐶𝐸 must have equal length. And since we’re given that 𝐷𝐴 is 69 centimeters, then 𝐶𝐸 must also have length 69 centimeters. We can also see that side 𝐸𝐵 of triangle 𝐴𝐵𝐸 has the same length as side 𝐶𝐸. So this too must have length 69 centimeters.
Now, we’re told in the question that the area of the trapezoid 𝐴𝐵𝐶𝐷 is 9522 square centimeters. And we know that the area of a trapezoid with vertical height ℎ, base 𝑏, and parallel side 𝑎 is given by 𝑎 plus 𝑏 over two multiplied by ℎ. Comparing to the given trapezoid, we can see that the base is equal to 𝐶𝐸 plus 𝐸𝐵. That’s 69 plus 69, which is 138. We see also that the side parallel to this is side 𝐷𝐴. So we have the parallel 𝑎 is our side 𝐷𝐴, and that’s 69 centimeters long.
We know the area, 9,522, and the only thing we don’t know in this formula for the area of trapezoid 𝐴𝐵𝐶𝐷 is the perpendicular height, ℎ. So, now to solve for ℎ, we can multiply both sides by two and evaluate inside the parentheses, giving 19,044 equals 207ℎ. Finally, dividing both sides by 207, we find ℎ equals 92 centimeters.
Now, in the given trapezoid, the height ℎ is actually the side length 𝐴𝐵. So we have side length 𝐴𝐵 equal to 92 centimeters. So, now clearing some space, we see that we have the lengths of two of the sides of triangle 𝐴𝐵𝐸. That’s 𝐴𝐵 equals 92 centimeters and 𝐸𝐵 equals 69 centimeters. And since this is a right triangle, we can use the Pythagorean theorem to find the third side 𝐴𝐸. We have 𝐴𝐵 squared plus 𝐸𝐵 squared equals 𝐴𝐸 squared. That’s 92 squared plus 69 squared equals 𝐴𝐸 squared.
Evaluating the left-hand side gives 13,225 equals 𝐴𝐸 squared. And taking the positive square root on both sides, positive since lengths are positive, we have 𝐴𝐸 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, it’s side 𝐵𝐹 that we want to find. And to use the corollary to find this, we need lengths 𝐴𝐹 and 𝐸𝐹. But since we now have side lengths 𝐴𝐵, 𝐸𝐵, and 𝐴𝐸, we can find these easily using the two formulae in the theorem.
In the first case, we have 92 squared, that’s 𝐴𝐵 squared, equals 𝐴𝐹 times 115, which is 𝐴𝐸. And dividing both sides by 115, then evaluating the left-hand side, we have 𝐴𝐹 equals 73.6. Now, for the second equation in the altitude theorem, we have 69 squared, that’s 𝐸𝐵 squared, equals 𝐸𝐹 times 115, which is 𝐸𝐴. Then, dividing both sides by 115 and evaluating, we have 𝐸𝐹 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 𝐵𝐹. This gives 𝐵𝐹 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 𝐵𝐹 equal to 55.2. Hence, using the right triangle altitude theorem and its corollary, we find the side length 𝐵𝐹 equals 55.2 centimeters.