# 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 π΄π΅ β line πΆπΉ, and the distance between them is β. Line segment π΄π· β line segment π΅πΆ, line segment π΄πΉ β line segment π΅πΈ. πΆπ· = 5 cm. And the area of parallelogram π΄π΅πΈπΉ is 15 cmΒ². Find β.

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### Video Transcript

In the opposite figure, line π΄π΅ is parallel to line πΆπΉ, and the distance between them is β. Line segment π΄π· is parallel to line segment π΅πΆ. Line segment π΄πΉ is parallel to line segment π΅πΈ. πΆπ· equals five centimeters. And the area of parallelogram π΄π΅πΈπΉ is 15 square centimeters. Find β.

We can begin this problem by noting that we have two parallel lines here with π΄π΅ and πΆπΉ. And the other two pairs of line segments, π΄π· and π΅πΆ and π΄πΉ and π΅πΈ, are also parallel. And this confirms that the two polygons π΄π΅πΈπΉ and π΄π΅πΆπ· are in fact both parallelograms because each shape is a quadrilateral with two pairs of opposite sides parallel. So given that the length of πΆπ· is five centimeters and the area of π΄π΅πΈπΉ is 15 square centimeters, we need to work out the value of β.

There are a few ways in which we could answer this question. But all of them will involve this important line segment: π΄π΅. Notice that this line segment π΄π΅ is common to both the parallelograms π΄π΅πΆπ· and π΄π΅πΈπΉ. 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.

To find the area of a parallelogram, we multiply the base times the perpendicular height. The area of parallelogram π΄π΅πΆπ· is 15 square centimeters. And the base can be taken as πΆπ·, which is five centimeters. We know that β is the perpendicular height of the parallelogram as we are told that the distance between the parallel lines is β. So that is a perpendicular distance. To calculate the value of β then, we divide both sides by five to give us that β equals three centimeters.

And previously, we mentioned that the line segment π΄π΅ is an important line segment for any method. Thatβs because, as an alternative method, we can recognize that in parallelogram π΄π΅πΆπ·, line segment π΄π΅ is opposite to line segment πΆπ·. As opposite sides in a parallelogram are congruent, then π΄π΅ has a length of five centimeters. We could then have used the given area of π΄π΅πΈπΉ as 15 square centimeters with the base π΄π΅ of five centimeters to work out that β is three centimeters.