In the opposite figure, line 𝐴𝐵 is parallel to line 𝐶𝐹, and the distance between
them is ℎ. Line segment 𝐴𝐷 is parallel to line segment 𝐵𝐶. Line segment 𝐴𝐹 is parallel to line segment 𝐵𝐸. 𝐶𝐷 equals five centimeters. And the area of parallelogram 𝐴𝐵𝐸𝐹 is 15 square centimeters. Find ℎ.
We can begin this problem by noting that we have two parallel lines here with 𝐴𝐵
and 𝐶𝐹. And the other two pairs of line segments, 𝐴𝐷 and 𝐵𝐶 and 𝐴𝐹 and 𝐵𝐸, are also
parallel. And this confirms that the two polygons 𝐴𝐵𝐸𝐹 and 𝐴𝐵𝐶𝐷 are in fact both
parallelograms because each shape is a quadrilateral with two pairs of opposite
sides parallel. So given that the length of 𝐶𝐷 is five centimeters and the area of 𝐴𝐵𝐸𝐹 is 15
square centimeters, we need to work out the value of ℎ.
There are a few ways in which we could answer this question. But all of them will involve this important line segment: 𝐴𝐵. Notice that this line segment 𝐴𝐵 is common to both the parallelograms 𝐴𝐵𝐶𝐷 and
𝐴𝐵𝐸𝐹. 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 ℎ.
To find the area of a parallelogram, we multiply the base times the perpendicular
height. The area of parallelogram 𝐴𝐵𝐶𝐷 is 15 square centimeters. And the base can be taken as 𝐶𝐷, which is five centimeters. We know that ℎ is the perpendicular height of the parallelogram as we are told that
the distance between the parallel lines is ℎ. So that is a perpendicular distance. To calculate the value of ℎ then, we divide both sides by five to give us that ℎ
equals three centimeters.
And previously, we mentioned that the line segment 𝐴𝐵 is an important line segment
for any method. That’s because, as an alternative method, we can recognize that in parallelogram
𝐴𝐵𝐶𝐷, line segment 𝐴𝐵 is opposite to line segment 𝐶𝐷. As opposite sides in a parallelogram are congruent, then 𝐴𝐵 has a length of five
centimeters. We could then have used the given area of 𝐴𝐵𝐸𝐹 as 15 square centimeters with the
base 𝐴𝐵 of five centimeters to work out that ℎ is three centimeters.