Video Transcript
๐น๐ท๐ธ๐ถ is a parallelogram, where
๐น and ๐ท are the midpoints of line segment ๐ด๐ต and line segment ๐ด๐ถ,
respectively, and ๐ถ๐ธ equals six centimeters. Determine the length of ๐ต๐ถ.
Letโs take the information weโre
given and add that in to our diagram. If ๐น๐ท๐ธ๐ถ is a parallelogram, the
opposite sides are parallel, meaning ๐น๐ท is parallel to ๐ถ๐ธ and ๐น๐ถ is parallel
to ๐ท๐ธ. We also know that ๐ถ๐ธ measures six
centimeters. But based on our parallelogram
properties, we also know that opposite side lengths will be equal. And that means that ๐ท๐น must also
measure six centimeters.
But our missing side length is
๐ต๐ถ. And so weโll need to be able to say
something else here. If ๐ถ๐ธ is parallel to ๐น๐ท, we can
also say that ๐ต๐ธ is parallel to ๐น๐ท. And if we can say that line segment
๐ต๐ถ is parallel to line segment ๐น๐ท, 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 ๐น๐ท creates two similar triangles. The smaller triangle, triangle
๐ด๐ท๐น, is similar to the larger triangle, triangle ๐ด๐ถ๐ต. And in similar triangles,
corresponding side lengths are proportional. The side length ๐ด๐ท on the smaller
triangle corresponds to the side length ๐ด๐ถ on the larger triangle. And that will have to be equal to
the smaller triangleโs line segment ๐น๐ท over the larger triangleโs line segment
๐ต๐ถ.
If we try to plug in the
information we know, we only end up filling in the side length ๐น๐ท, which is six
centimeters. But we can think carefully about
this midpoint ๐ท. The midpoint ๐ท divides line
segment ๐ด๐ถ in half. And so we can say that ๐ด๐ถ is
equal to two times ๐ด๐ท. What weโre 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 ๐น๐ท is equal to six
centimeters, we know that ๐ต๐ถ will have to be equal to 12 centimeters, as 12 is
twice six.