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Question Video: Finding the Perpendicular Height of a Parallelogram between Two Parallel Lines given the Area of Another Parallelogram Mathematics

In the opposite figure, line 𝐴𝐡 β€– line 𝐢𝐹, and the distance between them is β„Ž. Line segment 𝐴𝐷 β€– line segment 𝐡𝐢, line segment 𝐴𝐹 β€– line segment 𝐡𝐸. 𝐢𝐷 = 5 cm. And the area of parallelogram 𝐴𝐡𝐸𝐹 is 15 cmΒ². Find β„Ž.

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Video Transcript

In the opposite figure, line 𝐴𝐡 is parallel to line 𝐢𝐹, and the distance between them is β„Ž. Line segment 𝐴𝐷 is parallel to line segment 𝐡𝐢. Line segment 𝐴𝐹 is parallel to line segment 𝐡𝐸. 𝐢𝐷 equals five centimeters. And the area of parallelogram 𝐴𝐡𝐸𝐹 is 15 square centimeters. Find β„Ž.

We can begin this problem by noting that we have two parallel lines here with 𝐴𝐡 and 𝐢𝐹. And the other two pairs of line segments, 𝐴𝐷 and 𝐡𝐢 and 𝐴𝐹 and 𝐡𝐸, are also parallel. And this confirms that the two polygons 𝐴𝐡𝐸𝐹 and 𝐴𝐡𝐢𝐷 are in fact both parallelograms because each shape is a quadrilateral with two pairs of opposite sides parallel. So given that the length of 𝐢𝐷 is five centimeters and the area of 𝐴𝐡𝐸𝐹 is 15 square centimeters, we need to work out the value of β„Ž.

There are a few ways in which we could answer this question. But all of them will involve this important line segment: 𝐴𝐡. Notice that this line segment 𝐴𝐡 is common to both the parallelograms 𝐴𝐡𝐢𝐷 and 𝐴𝐡𝐸𝐹. And given that the parallelograms are between parallel lines, that should make us remember the theorem that parallelograms between a pair of parallel lines have the same area when their bases are the same length or when they share a common base. So both these parallelograms have the same area. They will both have an area of 15 square centimeters. And we can use this area to help us work out the value of β„Ž.

To find the area of a parallelogram, we multiply the base times the perpendicular height. The area of parallelogram 𝐴𝐡𝐢𝐷 is 15 square centimeters. And the base can be taken as 𝐢𝐷, which is five centimeters. We know that β„Ž is the perpendicular height of the parallelogram as we are told that the distance between the parallel lines is β„Ž. So that is a perpendicular distance. To calculate the value of β„Ž then, we divide both sides by five to give us that β„Ž equals three centimeters.

And previously, we mentioned that the line segment 𝐴𝐡 is an important line segment for any method. That’s because, as an alternative method, we can recognize that in parallelogram 𝐴𝐡𝐢𝐷, line segment 𝐴𝐡 is opposite to line segment 𝐢𝐷. As opposite sides in a parallelogram are congruent, then 𝐴𝐡 has a length of five centimeters. We could then have used the given area of 𝐴𝐡𝐸𝐹 as 15 square centimeters with the base 𝐴𝐡 of five centimeters to work out that β„Ž is three centimeters.

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