Video: Finding the Side Length of a Parallelogram Using the Similarity of Triangles

Given that 𝐴𝐡𝐢𝐷 is a parallelogram, 𝐡 is the midpoint of line segment 𝐴𝐹, 𝐢𝐸 = 8 cm, 𝐷𝐸 = 16 cm, and 𝐢𝑋 = 11 cm, find the length of line segment 𝐴𝐷.

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Video Transcript

Given that 𝐴𝐡𝐢𝐷 is a parallelogram, 𝐡 is the midpoint of line segment 𝐴𝐹, 𝐢𝐸 equals eight centimeters, 𝐷𝐸 equals 16 centimeters, and 𝐢𝑋 equals 11 centimeters, find the length of line segment 𝐴𝐷.

In this question, we’re told firstly that 𝐴𝐡𝐢𝐷 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 𝐢𝐡 and line 𝐢𝐷 is parallel to 𝐡𝐴. The next thing we’re told is that 𝐡 is the midpoint of line segment 𝐴𝐹. And we’re given the measurements that 𝐢𝐴 is eight centimeters, 𝐷𝐴 is 16 centimeters, and 𝐢𝑋 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 𝐢𝐡, 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 𝐢𝑋 and not to the entire line segment of 𝐢𝐡. So let’s see if there’s a way in which we could calculate this line segment 𝑋𝐡.

Let’s consider these two triangles. We have the smaller triangle 𝐢𝐸𝑋 and the larger triangle of 𝐡𝐹𝑋. 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 𝐸𝑋𝐢 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 𝐡𝑋𝐹. So these two angles will be equal in size.

Next, let’s look at the angle 𝐸𝐢𝑋. If we consider that we have the parallel lines 𝐷𝐢 and 𝐴𝐹, then the transversal of 𝐢𝐡 would give us an equal angle. We can say that angle 𝐢𝐡𝐹 is equal to the angle 𝐴𝐢𝑋 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, 𝐢𝐸𝑋 and 𝐡𝐹𝑋, 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 𝑋𝐡.

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 𝐡𝐹, but we can in fact work it out relatively simply. Because we’re told that 𝐡 is the midpoint of 𝐴𝐹, then the length of the line segment 𝐡𝐹 will be the same as the length of this line segment 𝐴𝐡.

Using the fact that, in a parallelogram, opposite sides are parallel and congruent, then the length of 𝐴𝐡 will be equal to the sum of eight centimeters and 16 centimeters. And that’s 24 centimeters. And so, 𝐡𝐹 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 𝐢𝐴 is eight centimeters and 𝐡𝐹 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 𝐡𝑋, so which side on the smaller triangle would be corresponding with this length? It would be this one, the line segment 𝐢𝑋. We can therefore take 𝐢𝑋 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.

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