In the figure below, line 𝐴𝐵 is
parallel to line 𝐶𝐸, line segment 𝐴𝐶 is parallel to line segment 𝐵𝐷, and
𝐴𝐵𝐸𝐹 is a rectangle. If 𝐵𝐸 equals four centimeters and
𝐴𝐵 equals three centimeters, find the area of parallelogram 𝐴𝐵𝐷𝐶.
From the information that we are
given, we note that we have two pairs of parallel lines, which confirms that
𝐴𝐵𝐷𝐶 is indeed a parallelogram. We are further told that 𝐴𝐵𝐸𝐹
is a rectangle. Since a rectangle is simply a
special case of parallelogram, it means we can also note that the line segments 𝐴𝐹
and 𝐵𝐸 are also parallel. We can use the information about
𝐴𝐵𝐸𝐹 to help us work out the area of the parallelogram 𝐴𝐵𝐷𝐶. We can use the information that
𝐵𝐸 is equal to four centimeters and 𝐴𝐵 is equal to three centimeters to help us
work out the area of the rectangle 𝐴𝐵𝐸𝐹.
If you’re not sure why this is
useful, let’s recall an important property about parallelograms created between two
parallel lines. Parallelograms between a pair of
parallel lines with congruent bases have the same area, so even though 𝐴𝐵𝐸𝐹 is a
rectangle, that’s a special type of parallelogram, and the line segment 𝐴𝐵 is a
common side to both 𝐴𝐵𝐸𝐹 and 𝐴𝐵𝐷𝐶. So if we work out the area of
𝐴𝐵𝐸𝐹, it’s going to be the same as the area of 𝐴𝐵𝐷𝐶.
And of course, to work out the area
of a rectangle, we multiply the length by the width. Three times four is 12, and the
area units will be square centimeters. The area of 𝐴𝐵𝐷𝐶 is going to be
equal to this. So it’s also 12 square
centimeters. We could also have worked out the
area of 𝐴𝐵𝐷𝐶 directly. The area of a parallelogram is
found by multiplying the base by the perpendicular height. The base of 𝐴𝐵𝐷𝐶 is three
centimeters, and the perpendicular height is also the length of the line segment
𝐵𝐸, which is four centimeters. Either method would produce the
result that the area of 𝐴𝐵𝐷𝐶 is 12 square centimeters.