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

In the figure below, line 𝐴𝐡 ⫽ line 𝐢𝐸, line segment 𝐴𝐢 ⫽ line segment 𝐡𝐷, and 𝐴𝐡𝐸𝐹 is a rectangle. If 𝐡𝐸 = 4 cm and 𝐴𝐡 = 3 cm, find the area of parallelogram 𝐴𝐡𝐷𝐢.

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Video Transcript

In the figure below, line 𝐴𝐡 is parallel to line 𝐢𝐸, line segment 𝐴𝐢 is parallel to line segment 𝐡𝐷, and 𝐴𝐡𝐸𝐹 is a rectangle. If 𝐡𝐸 equals four centimeters and 𝐴𝐡 equals three centimeters, find the area of parallelogram 𝐴𝐡𝐷𝐢.

From the information that we are given, we note that we have two pairs of parallel lines, which confirms that 𝐴𝐡𝐷𝐢 is indeed a parallelogram. We are further told that 𝐴𝐡𝐸𝐹 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 𝐡𝐸 are also parallel. We can use the information about 𝐴𝐡𝐸𝐹 to help us work out the area of the parallelogram 𝐴𝐡𝐷𝐢. We can use the information that 𝐡𝐸 is equal to four centimeters and 𝐴𝐡 is equal to three centimeters to help us work out the area of the rectangle 𝐴𝐡𝐸𝐹.

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 𝐴𝐡𝐸𝐹 is a rectangle, that’s a special type of parallelogram, and the line segment 𝐴𝐡 is a common side to both 𝐴𝐡𝐸𝐹 and 𝐴𝐡𝐷𝐢. So if we work out the area of 𝐴𝐡𝐸𝐹, it’s going to be the same as the area of 𝐴𝐡𝐷𝐢.

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 𝐴𝐡𝐷𝐢 is going to be equal to this. So it’s also 12 square centimeters. We could also have worked out the area of 𝐴𝐡𝐷𝐢 directly. The area of a parallelogram is found by multiplying the base by the perpendicular height. The base of 𝐴𝐡𝐷𝐢 is three centimeters, and the perpendicular height is also the length of the line segment 𝐡𝐸, which is four centimeters. Either method would produce the result that the area of 𝐴𝐡𝐷𝐢 is 12 square centimeters.

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