### Video Transcript

πΉπ·πΈπΆ is a parallelogram, where
πΉ and π· are the midpoints of line segment π΄π΅ and line segment π΄πΆ,
respectively, and πΆπΈ equals six centimeters. Determine the length of π΅πΆ.

Letβs take the information weβre
given and add that in to our diagram. If πΉπ·πΈπΆ is a parallelogram, the
opposite sides are parallel, meaning πΉπ· is parallel to πΆπΈ and πΉπΆ is parallel
to π·πΈ. We also know that πΆπΈ measures six
centimeters. But based on our parallelogram
properties, we also know that opposite side lengths will be equal. And that means that π·πΉ must also
measure six centimeters.

But our missing side length is
π΅πΆ. And so weβll need to be able to say
something else here. If πΆπΈ is parallel to πΉπ·, we can
also say that π΅πΈ is parallel to πΉπ·. And if we can say that line segment
π΅πΆ is parallel to line segment πΉπ·, 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 πΉπ· creates two similar triangles. The smaller triangle, triangle
π΄π·πΉ, is similar to the larger triangle, triangle π΄πΆπ΅. And in similar triangles,
corresponding side lengths are proportional. The side length π΄π· on the smaller
triangle corresponds to the side length π΄πΆ on the larger triangle. And that will have to be equal to
the smaller triangleβs line segment πΉπ· over the larger triangleβs line segment
π΅πΆ.

If we try to plug in the
information we know, we only end up filling in the side length πΉπ·, which is six
centimeters. But we can think carefully about
this midpoint π·. The midpoint π· divides line
segment π΄πΆ in half. And so we can say that π΄πΆ is
equal to two times π΄π·. What weβre 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 πΉπ· is equal to six
centimeters, we know that π΅πΆ will have to be equal to 12 centimeters, as 12 is
twice six.