Video Transcript
Given that ๐ด๐ต๐ถ๐ท is a parallelogram, ๐ต is the midpoint of line segment ๐ด๐น, ๐ถ๐ธ
equals eight centimeters, ๐ท๐ธ equals 16 centimeters, and ๐ถ๐ equals 11
centimeters, find the length of line segment ๐ด๐ท.
In this question, weโre told firstly that ๐ด๐ต๐ถ๐ท is a parallelogram. Thatโs the quadrilateral at the top of this figure. One of the main properties that we should remember about parallelograms are that
opposite sides are parallel. In the figure, this would mean that line ๐ท๐ด is parallel to line ๐ถ๐ต and line ๐ถ๐ท
is parallel to ๐ต๐ด. The next thing weโre told is that ๐ต is the midpoint of line segment ๐ด๐น. And weโre given the measurements that ๐ถ๐ด is eight centimeters, ๐ท๐ด is 16
centimeters, and ๐ถ๐ equals 11 centimeters, so we can fill in these measurements
onto the diagram.
The question asks us to find the length of this line segment ๐ด๐ท. In order to work this out, we should recall another important fact about the opposite
sides in a parallelogram. We know that theyโre parallel, but we should also recall that opposite sides are
congruent. That means theyโre the same length. Therefore, if we need a length of the opposite side to ๐ด๐ท, the length of the line
segment ๐ถ๐ต, then we would know that it would be of the same length. The length of 11 centimeters that weโre given here only refers to the line segment
๐ถ๐ and not to the entire line segment of ๐ถ๐ต. So letโs see if thereโs a way in which we could calculate this line segment ๐๐ต.
Letโs consider these two triangles. We have the smaller triangle ๐ถ๐ธ๐ and the larger triangle of ๐ต๐น๐. We might then ask ourselves if these two triangles are similar. That means theyโll be the same shape but a different size. One way that we can prove similarity in triangles is by using the AA criterion in
which we see if there are two pairs of corresponding pairs of angles congruent. So letโs check out the angles in these triangles. Letโs look at this angle ๐ธ๐๐ถ and see if there would be an equal angle to this
one. Well, in fact, there would. Thereโs a vertically opposite angle, the angle ๐ต๐๐น. So these two angles will be equal in size.
Next, letโs look at the angle ๐ธ๐ถ๐. If we consider that we have the parallel lines ๐ท๐ถ and ๐ด๐น, then the transversal of
๐ถ๐ต would give us an equal angle. We can say that angle ๐ถ๐ต๐น is equal to the angle ๐ด๐ถ๐ because we have alternate
angles. As weโve demonstrated that there are two pairs of corresponding angles equal, then
weโve shown that the AA rule is satisfied. This means that our two triangles, ๐ถ๐ธ๐ and ๐ต๐น๐, are indeed similar. So letโs return to the reason that we wanted to check these two triangles were
similar, and thatโs to find the length of the line segment ๐๐ต.
In these similar triangles, theyโll be the same proportion between every length on
the smaller triangle to every length on the larger triangle. We could also think of this in terms of finding the scale factor. In order to find this scale factor, we need a corresponding pair of lengths on the
smaller triangle and on the larger triangle. Now, weโre not given the length of this line segment ๐ต๐น, but we can in fact work it
out relatively simply. Because weโre told that ๐ต is the midpoint of ๐ด๐น, then the length of the line
segment ๐ต๐น will be the same as the length of this line segment ๐ด๐ต.
Using the fact that, in a parallelogram, opposite sides are parallel and congruent,
then the length of ๐ด๐ต will be equal to the sum of eight centimeters and 16
centimeters. And thatโs 24 centimeters. And so, ๐ต๐น is also 24 centimeters. We now have a pair of corresponding sides in the smaller triangle and the larger
triangle, which would allow us to work out the scale factor. If ๐ถ๐ด is eight centimeters and ๐ต๐น is 24 centimeters, then the scale factor must
be three. That means each length on the smaller triangle could be multiplied by three to give
the corresponding length on the larger triangle.
We remember that weโre trying to find this length of the line segment ๐ต๐, so which
side on the smaller triangle would be corresponding with this length? It would be this one, the line segment ๐ถ๐. We can therefore take ๐ถ๐ of 11 centimeters and multiply it by the scale factor of
three to give us a length of 33 centimeters. And finally, after all of that working out, weโre about ready to calculate the length
of the line segment ๐ด๐ท. Remembering that the parallelogram has opposite sides congruent, then weโre adding 11
centimeters and 33 centimeters, which gives us the final answer that the length of
line segment ๐ด๐ท is 44 centimeters.