### Video Transcript

In the opposite figure, line π΄π΅ is parallel to line πΆπΉ, and the distance between
them is β. Line segment π΄π· is parallel to line segment π΅πΆ. Line segment π΄πΉ is parallel to line segment π΅πΈ. πΆπ· equals five centimeters. And the area of parallelogram π΄π΅πΈπΉ is 15 square centimeters. Find β.

We can begin this problem by noting that we have two parallel lines here with π΄π΅
and πΆπΉ. And the other two pairs of line segments, π΄π· and π΅πΆ and π΄πΉ and π΅πΈ, are also
parallel. And this confirms that the two polygons π΄π΅πΈπΉ and π΄π΅πΆπ· are in fact both
parallelograms because each shape is a quadrilateral with two pairs of opposite
sides parallel. So given that the length of πΆπ· is five centimeters and the area of π΄π΅πΈπΉ 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΅πΆπ· and
π΄π΅πΈπΉ. 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΅πΆπ· is 15 square centimeters. And the base can be taken as πΆπ·, 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. Thatβs because, as an alternative method, we can recognize that in parallelogram
π΄π΅πΆπ·, line segment π΄π΅ is opposite to line segment πΆπ·. 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΅πΈπΉ as 15 square centimeters with the
base π΄π΅ of five centimeters to work out that β is three centimeters.