Question Video: Finding the Area of a Parallelogram Using the Area of a Rectangle Mathematics

In the figure below, line π΄π΅ β«½ line πΆπΈ, line segment π΄πΆ β«½ line segment π΅π·, and π΄π΅πΈπΉ is a rectangle. If π΅πΈ = 4 cm and π΄π΅ = 3 cm, find the area of parallelogram π΄π΅π·πΆ.

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Video Transcript

In the figure below, line π΄π΅ is parallel to line πΆπΈ, line segment π΄πΆ is parallel to line segment π΅π·, and π΄π΅πΈπΉ is a rectangle. If π΅πΈ equals four centimeters and π΄π΅ equals three centimeters, find the area of parallelogram π΄π΅π·πΆ.

From the information that we are given, we note that we have two pairs of parallel lines, which confirms that π΄π΅π·πΆ is indeed a parallelogram. We are further told that π΄π΅πΈπΉ is a rectangle. Since a rectangle is simply a special case of parallelogram, it means we can also note that the line segments π΄πΉ and π΅πΈ are also parallel. We can use the information about π΄π΅πΈπΉ to help us work out the area of the parallelogram π΄π΅π·πΆ. We can use the information that π΅πΈ is equal to four centimeters and π΄π΅ is equal to three centimeters to help us work out the area of the rectangle π΄π΅πΈπΉ.

If youβre not sure why this is useful, letβ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 π΄π΅πΈπΉ is a rectangle, thatβs a special type of parallelogram, and the line segment π΄π΅ is a common side to both π΄π΅πΈπΉ and π΄π΅π·πΆ. So if we work out the area of π΄π΅πΈπΉ, itβs going to be the same as the area of π΄π΅π·πΆ.

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 π΄π΅π·πΆ is going to be equal to this. So itβs also 12 square centimeters. We could also have worked out the area of π΄π΅π·πΆ directly. The area of a parallelogram is found by multiplying the base by the perpendicular height. The base of π΄π΅π·πΆ is three centimeters, and the perpendicular height is also the length of the line segment π΅πΈ, which is four centimeters. Either method would produce the result that the area of π΄π΅π·πΆ is 12 square centimeters.