Video: Finding the Unknown Lengths in a Triangle given the Other Sides’ Lengths Using the Relations of Parallel Lines

Video Transcript

𝐹𝐷𝐸𝐶 is a parallelogram, where 𝐹 and 𝐷 are the midpoints of line segment 𝐴𝐵 and line segment 𝐴𝐶, respectively, and 𝐶𝐸 equals six centimeters. Determine the length of 𝐵𝐶.

Let’s take the information we’re given and add that in to our diagram. If 𝐹𝐷𝐸𝐶 is a parallelogram, the opposite sides are parallel, meaning 𝐹𝐷 is parallel to 𝐶𝐸 and 𝐹𝐶 is parallel to 𝐷𝐸. We also know that 𝐶𝐸 measures six centimeters. But based on our parallelogram properties, we also know that opposite side lengths will be equal. And that means that 𝐷𝐹 must also measure six centimeters.

But our missing side length is 𝐵𝐶. And so we’ll need to be able to say something else here. If 𝐶𝐸 is parallel to 𝐹𝐷, we can also say that 𝐵𝐸 is parallel to 𝐹𝐷. And if we can say that line segment 𝐵𝐶 is parallel to line segment 𝐹𝐷, then we have a line that is parallel to a side of our triangle. And that line intersects the other two sides, which means the line segment 𝐹𝐷 creates two similar triangles. The smaller triangle, triangle 𝐴𝐷𝐹, is similar to the larger triangle, triangle 𝐴𝐶𝐵. And in similar triangles, corresponding side lengths are proportional. The side length 𝐴𝐷 on the smaller triangle corresponds to the side length 𝐴𝐶 on the larger triangle. And that will have to be equal to the smaller triangle’s line segment 𝐹𝐷 over the larger triangle’s line segment 𝐵𝐶.

If we try to plug in the information we know, we only end up filling in the side length 𝐹𝐷, which is six centimeters. But we can think carefully about this midpoint 𝐷. The midpoint 𝐷 divides line segment 𝐴𝐶 in half. And so we can say that 𝐴𝐶 is equal to two times 𝐴𝐷. What we’re saying is the ratio of the smaller triangle to the larger triangle would be one-half since the larger triangle is always two times greater than the smaller triangle. And if the side lengths of the larger triangle is two times that of the smaller triangle and 𝐹𝐷 is equal to six centimeters, we know that 𝐵𝐶 will have to be equal to 12 centimeters, as 12 is twice six.