Given that 𝐴𝐵𝐶𝐷 is a parallelogram, 𝐵 is the midpoint of line segment 𝐴𝐹, 𝐶𝐸
equals eight centimeters, 𝐷𝐸 equals 16 centimeters, and 𝐶𝑋 equals 11
centimeters, find the length of line segment 𝐴𝐷.
In this question, we’re told firstly that 𝐴𝐵𝐶𝐷 is a parallelogram. That’s 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 𝐷𝐴 is parallel to line 𝐶𝐵 and line 𝐶𝐷
is parallel to 𝐵𝐴. The next thing we’re told is that 𝐵 is the midpoint of line segment 𝐴𝐹. And we’re given the measurements that 𝐶𝐴 is eight centimeters, 𝐷𝐴 is 16
centimeters, and 𝐶𝑋 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 𝐴𝐷. In order to work this out, we should recall another important fact about the opposite
sides in a parallelogram. We know that they’re parallel, but we should also recall that opposite sides are
congruent. That means they’re the same length. Therefore, if we need a length of the opposite side to 𝐴𝐷, the length of the line
segment 𝐶𝐵, then we would know that it would be of the same length. The length of 11 centimeters that we’re given here only refers to the line segment
𝐶𝑋 and not to the entire line segment of 𝐶𝐵. So let’s see if there’s a way in which we could calculate this line segment 𝑋𝐵.
Let’s consider these two triangles. We have the smaller triangle 𝐶𝐸𝑋 and the larger triangle of 𝐵𝐹𝑋. We might then ask ourselves if these two triangles are similar. That means they’ll 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 let’s check out the angles in these triangles. Let’s look at this angle 𝐸𝑋𝐶 and see if there would be an equal angle to this
one. Well, in fact, there would. There’s a vertically opposite angle, the angle 𝐵𝑋𝐹. So these two angles will be equal in size.
Next, let’s look at the angle 𝐸𝐶𝑋. If we consider that we have the parallel lines 𝐷𝐶 and 𝐴𝐹, then the transversal of
𝐶𝐵 would give us an equal angle. We can say that angle 𝐶𝐵𝐹 is equal to the angle 𝐴𝐶𝑋 because we have alternate
angles. As we’ve demonstrated that there are two pairs of corresponding angles equal, then
we’ve shown that the AA rule is satisfied. This means that our two triangles, 𝐶𝐸𝑋 and 𝐵𝐹𝑋, are indeed similar. So let’s return to the reason that we wanted to check these two triangles were
similar, and that’s to find the length of the line segment 𝑋𝐵.
In these similar triangles, they’ll 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, we’re not given the length of this line segment 𝐵𝐹, but we can in fact work it
out relatively simply. Because we’re told that 𝐵 is the midpoint of 𝐴𝐹, then the length of the line
segment 𝐵𝐹 will be the same as the length of this line segment 𝐴𝐵.
Using the fact that, in a parallelogram, opposite sides are parallel and congruent,
then the length of 𝐴𝐵 will be equal to the sum of eight centimeters and 16
centimeters. And that’s 24 centimeters. And so, 𝐵𝐹 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 𝐶𝐴 is eight centimeters and 𝐵𝐹 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 we’re trying to find this length of the line segment 𝐵𝑋, so which
side on the smaller triangle would be corresponding with this length? It would be this one, the line segment 𝐶𝑋. We can therefore take 𝐶𝑋 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, we’re about ready to calculate the length
of the line segment 𝐴𝐷. Remembering that the parallelogram has opposite sides congruent, then we’re adding 11
centimeters and 33 centimeters, which gives us the final answer that the length of
line segment 𝐴𝐷 is 44 centimeters.