Question Video: Finding the Area of a Triangle between Parallel Lines Mathematics

Video Transcript

In the figure shown, the line between 𝐷 and 𝐶 is parallel to the line between 𝐴 and 𝐹, the line segment between 𝐴 and 𝐷 is parallel to the line segment between 𝐵 and 𝐶, and the line segment between 𝐷 and 𝐸 is parallel to the line between 𝐶 and 𝐹. If the area of parallelogram 𝐴𝐵𝐶𝐷 is 16 square centimeters, find the area of triangle 𝐸𝐹𝐺.

In this question, we are given three pairs of parallel lines and the area of a parallelogram and asked to determine the area of a triangle. We can begin by adding the pairs of parallel lines to the given diagram. We can also highlight the parallelogram 𝐴𝐵𝐶𝐷 whose area is 16 square centimeters on the given diagram as shown. We want to find the area of triangle 𝐸𝐹𝐺, which we can highlight as shown. Since both the parallelogram and triangle have bases on the same pair of parallel lines and vertices on the other parallel line, they have the same perpendicular height. We can use this idea to find the area of triangle 𝐸𝐹𝐺 from the area of the given parallelogram.

To begin, we see that parallelogram 𝐶𝐷𝐸𝐹 highlighted in green shares base 𝐶𝐷 with parallelogram 𝐴𝐵𝐶𝐷 and they are between the same pair of parallel lines. So they have the same perpendicular height. Since the area of a parallelogram is the length of the base times the perpendicular height and these parallelograms have the same base length and perpendicular height, they must have the same area. So, the area of parallelogram 𝐶𝐷𝐸𝐹 is also 16 square centimeters.

We then see that triangle 𝐸𝐹𝐺 has the same perpendicular height as parallelogram 𝐶𝐷𝐸𝐹, and they share the same base 𝐸𝐹. This allows us to find the area of triangle 𝐸𝐹𝐺 by recalling that its area is one-half the length of the base times the perpendicular height, which is exactly the same as one-half the area of parallelogram 𝐶𝐷𝐸𝐹. Hence, the area of triangle 𝐸𝐹𝐺 is one-half times 16, which is equal to eight square centimeters.