Question Video: Finding the Area of a Parallelogram given the Area of a Triangle That Shares the Same Base Mathematics • Second Year of Preparatory School

Video Transcript

In the opposite figure, line 𝐴𝐵 is parallel to line 𝐸𝐷 and line segment 𝐴𝐶 is parallel to line segment 𝐵𝐷. If the area of triangle 𝐴𝐵𝐸 equals seven square centimeters, find the area of parallelogram 𝐴𝐵𝐷𝐶.

We can observe here that since the two lines 𝐴𝐵 and 𝐸𝐷 are parallel and the two line segments 𝐴𝐶 and 𝐵𝐷 are parallel, then we have indeed got a parallelogram in 𝐴𝐵𝐷𝐶. Given that the area of triangle 𝐴𝐵𝐸 is seven square centimeters, we need to work out the area of parallelogram 𝐴𝐵𝐷𝐶. To do this, we can use the fact that the triangle and the parallelogram share the same base of 𝐴𝐵. Importantly, they also share the same perpendicular height, which we can define with the letter ℎ.

Now to find the area of a triangle, we recall that this is equal to one-half multiplied by the base multiplied by the perpendicular height. So the area of triangle 𝐴𝐵𝐸 is equal to one-half multiplied by the length of the line segment 𝐴𝐵 multiplied by ℎ. Equally, we can recall that to find the area of a parallelogram, we multiply the base by the perpendicular height. Therefore, the area of parallelogram 𝐴𝐵𝐷𝐶 can be calculated by the length of 𝐴𝐵 multiplied by ℎ.

If we then observe the right-hand side of both of these equations, we can observe that the parallelogram is simply double the area of the triangle. Therefore, given that the area of triangle 𝐴𝐵𝐸 is seven square centimeters, we double it, giving us the answer that the area of parallelogram 𝐴𝐵𝐷𝐶 is 14 square centimeters.