![]() ![]() So, now taking the positive square root on both sides, we have □□ equal to 55.2. ![]() Now we can use these lengths in the corollary to find the side length □□. The sum does equal 115, so we can assume our side lengths are correct. We can check our side lengths are correct by summing our two answers. Then, dividing both sides by 115 and evaluating, we have □□ equal to 41.4. Now, for the second equation in the altitude theorem, we have 69 squared, that’s □□ squared, equals □□ times 115, which is □□. And dividing both sides by 115, then evaluating the left-hand side, we have □□ equals 73.6. In the first case, we have 92 squared, that’s □□ squared, equals □□ times 115, which is □□. But since we now have side lengths □□, □□, and □□, we can find these easily using the two formulae in the theorem. And to use the corollary to find this, we need lengths □□ and □□. Remember, it’s side □□ that we want to find. Marking this on our diagram, we can now turn our attention to the right triangle altitude theorem and its corollary. ![]() And taking the positive square root on both sides, positive since lengths are positive, we have □□ equal to 115 centimeters. That’s 92 squared plus 69 squared equals □□ squared.Įvaluating the left-hand side gives 13,225 equals □□ squared. We have □□ squared plus □□ squared equals □□ squared. And since this is a right triangle, we can use the Pythagorean theorem to find the third side □□. That’s □□ equals 92 centimeters and □□ equals 69 centimeters. So, now clearing some space, we see that we have the lengths of two of the sides of triangle □□□. So we have side length □□ equal to 92 centimeters. Now, in the given trapezoid, the height ℎ is actually the side length □□. Finally, dividing both sides by 207, we find ℎ equals 92 centimeters. So, now to solve for ℎ, we can multiply both sides by two and evaluate inside the parentheses, giving 19,044 equals 207ℎ. We know the area, 9,522, and the only thing we don’t know in this formula for the area of trapezoid □□□□ is the perpendicular height, ℎ. So we have the parallel □ is our side □□, and that’s 69 centimeters long. We see also that the side parallel to this is side □□. Comparing to the given trapezoid, we can see that the base is equal to □□ plus □□. ![]() And we know that the area of a trapezoid with vertical height ℎ, base □, and parallel side □ is given by □ plus □ over two multiplied by ℎ. Now, we’re told in the question that the area of the trapezoid □□□□ is 9522 square centimeters. So this too must have length 69 centimeters. We can also see that side □□ of triangle □□□ has the same length as side □□. And since we’re given that □□ is 69 centimeters, then □□ must also have length 69 centimeters. This means that sides □□ and □□ must have equal length. We can therefore conclude that □□□□ is a parallelogram within the trapezoid □□□□. So we can see that sides □□ and □□ are parallel, as are sides □□ and □□. And for this, we first need to extract some information from the diagram. To do all this, we’re going to need to find each of the side lengths of triangle □□□. And taking the square root will give us the length of □□. And to find this length, we’re going to use the right triangle altitude theorem to find the lengths of sides □□ and □□ and then its corollary to find the square of the side we want, which is side □□. In this example, we want to find □□, which is the perpendicular projection to □, from the right angle at □ in triangle □□□. Given that the area of the trapezoid □□□□ is 9,522 square centimeters, determine the length of the side □□. ![]()
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