Video Transcript
In the opposite figure, 𝐴𝐵𝐶𝐷
and 𝐴𝐵𝐸𝐹 are two parallelograms with common base of line segment 𝐴𝐵. Line segment 𝐴𝐵 is parallel to
line segment 𝐶𝐹 and the distance between them is ℎ, where ℎ equals five
centimeters and 𝐶𝐷 equals four centimeters. Find the area of parallelogram
𝐴𝐵𝐸𝐹.
Let’s begin this question by
filling in the information given in the question onto the diagram. The lines 𝐴𝐵 and 𝐶𝐹 are
parallel, and the distance between these is ℎ. We are also told that ℎ is five
centimeters. Then, we are given that 𝐶𝐷 equals
four centimeters. Now, let’s consider the two
parallelograms 𝐴𝐵𝐶𝐷 and 𝐴𝐵𝐸𝐹. We are asked to find the area of
parallelogram 𝐴𝐵𝐸𝐹. Because we are told that the two
parallelograms share the same base of line segment 𝐴𝐵, we can recall an important
theorem. And that is that parallelograms
between a pair of parallel lines have the same area when they share a common
base. This is also true if the base is
the same length, even if it is not common. So the area of parallelogram
𝐴𝐵𝐸𝐹 equals the area of parallelogram 𝐴𝐵𝐶𝐷.
Let’s remind ourselves of the
formula to find the area of a parallelogram. The area of a parallelogram is
equal to the base times the perpendicular height. So let’s consider if we have this
information for either 𝐴𝐵𝐸𝐹 or 𝐴𝐵𝐶𝐷. In 𝐴𝐵𝐶𝐷, the base length of
𝐶𝐷 was given as four centimeters. And the perpendicular height is
five centimeters. We know that this is the
perpendicular height because we were told that ℎ is the distance between the
parallel lines 𝐴𝐵 and 𝐶𝐹. So to work out the area of
parallelogram 𝐴𝐵𝐶𝐷, we can multiply the base of four by the perpendicular height
of five centimeters, which gives 20 square centimeters. And since we know that the two
parallelograms have equal areas, then the area of 𝐴𝐵𝐸𝐹 is also 20 square
centimeters.
Note that we could also have
calculated this using the fact that in a parallelogram opposite sides are
congruent. And therefore, since 𝐶𝐷 has a
length of four centimeters, then 𝐴𝐵 will have the same length. And then we could have directly
calculated the area of 𝐴𝐵𝐸𝐹 as four times five, which is 20 square
centimeters. But the equality of the areas of
parallelograms between parallel lines is a very useful theorem to be able to recall
and use.
