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

Video Transcript

In the opposite figure, the line passing through the points 𝐴 and 𝐵 is parallel to the line passing through the points 𝐶 and 𝐷. The line segment 𝐴𝐶 is parallel to the line segment 𝐵𝐷. And the line segment 𝐹𝐺 is perpendicular to the line segment 𝐴𝐵, where the length 𝐹𝐺 is equal to four centimeters. If the area of the triangle 𝐴𝐵𝐸 is equal to 12 centimeters squared, find the length of the line segment 𝐶𝐷.

Let’s start with labeling the diagram with the information given. The length of 𝐹𝐺 is four centimeters, and the area of the triangle 𝐴𝐵𝐸 is 12 centimeters squared. Recall that the area of a triangle is given by one-half times its base, 𝑏, times its perpendicular height, ℎ.

Looking at the triangle 𝐴𝐵𝐸, we can take its base length to be the length of the line segment 𝐴𝐵. The height of the triangle is the distance between the top line and the point 𝐸, which is equivalent to the distance between the top and bottom lines. Since these lines are parallel and the line segment 𝐹𝐺 is perpendicular to them, the distance between both lines is just the length 𝐹𝐺. Therefore, the area of the triangle is one-half times 𝐴𝐵 times 𝐹𝐺, which we are given as equal to 12 centimeters squared.

We are also given that the length 𝐹𝐺 is equal to four centimeters. So on the left-hand side of this equation, we have one-half times 𝐴𝐵 times four, which is equal to two 𝐴𝐵. So we have two 𝐴𝐵 equals 12. Dividing both sides by two gives us the length of the line segment 𝐴𝐵, six centimeters.

Now consider the quadrilateral 𝐴𝐵𝐶𝐷. Since, as given in the question, the line segments 𝐴𝐵 and 𝐶𝐷 are parallel, the line segments 𝐴𝐶 and 𝐵𝐷 are also parallel. So this is a parallelogram, and the parallel sides are of equal length. Therefore, the length of the line segment 𝐶𝐷 is equal to the length of the line segment 𝐴𝐵, which is six centimeters.