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.
