Given that 𝐴𝐵𝐶𝐷 is similar to 𝑍𝑌𝑋𝐿, find the measure of angle 𝑋𝐿𝑍 and the length of line segment 𝐶𝐷.
The first thing to notice in this question is this curly line which indicates similarity. When we have two polygons which are similar, that means that they have corresponding angles equal and corresponding sides in proportion. What we need to do is to work out which pairs of angles and which pairs of sides will be corresponding. So the first part of this question asks us to find the measure of angle 𝑋𝐿𝑍, which will be here on the larger polygon. The angle on the smaller polygon 𝐴𝐵𝐶𝐷 that’s corresponding to 𝑋𝐿𝑍 will be this angle at 𝐶𝐷𝐴.
On some diagrams, it’s not easy to recognize which angles are corresponding, but we can always do it from the similarity statement if we’re given one. For example, in our similarity statement, angle 𝑋𝐿𝑍 will be corresponding with angle 𝐶𝐷𝐴. But here the problem is we don’t actually know the angle at 𝐶𝐷𝐴, so let’s see if we can find another pair of corresponding angles. We might notice that there’s this angle of 85 degrees in the smaller polygon; it’s the angle 𝐷𝐶𝐵. And the corresponding angle in the larger polygon will be the angle 𝐿𝑋𝑌. So the angle 𝐿𝑋𝑌 is also 85 degrees.
Now that we have three angles in this quadrilateral, we can find the measure of angle 𝑋𝐿𝑍 by remembering a key fact about the angles in a quadrilateral. The angles in a quadrilateral sum to 360 degrees. Therefore, if we add together the three angles of 105 degrees, 109 degrees, and 85 degrees and then subtract that from 360 degrees, we’ll find the measure of angle 𝑋𝐿𝑍. Calculating 360 degrees subtract 299 degrees gives us the angle of 61 degrees. And so that’s the answer for the first part of the question to find the measure of angle 𝑋𝐿𝑍.
We can now clear some space to find the length of the line segment 𝐶𝐷. As we’re looking at the lengths of the sides in these two similar polygons, it’s important to remember that corresponding sides are in proportion but not necessarily equal. To start looking at corresponding sides, we can observe that this length of 𝐴𝐵 is corresponding with the length of 𝑍𝑌. Usually, when we’re working with similar polygons, we’ll be given or could very easily work out a pair of corresponding lengths. Having a pair of corresponding sides allows us to work out the proportion between the shapes. There are two different methods that we can use to find the length of this line segment 𝐶𝐷, so let’s look at the first method.
In this method, we would say that there’s a proportion between the two corresponding sides of 𝑍𝑌 and 𝐴𝐵. And it’s going to be equal to the ratio or proportion between the line segment 𝐶𝐷 and its corresponding side. The side that’s corresponding to 𝐶𝐷 will be this length of 𝑋𝐿. So we’re really saying that the proportion of 𝑋𝑌 to 𝐴𝐵 is the same as the proportion of 𝑋𝐿 to 𝐶𝐷.
Now all we need to do is fill in the values that we’re given for each of these lengths. So we have that 150 over 75 is equal to 246.2 over 𝐶𝐷, which we’re trying to calculate. We can simplify the fraction on the left-hand side by taking out a common factor of 75. Taking the cross product, we find that two times the length of 𝐶𝐷 is equal to 246.2 multiplied by one, which is just 246.2. Dividing both sides of this equation by two, we find that the length of line segment 𝐶𝐷 is 123.1, and of course the units will be the length units of centimeters. And so that’s the answer for the second part of this question.
But before we finish, let’s have a look at the second alternative method we could’ve used to find the length of line segment 𝐶𝐷. And it involves finding the scale factor between the two polygons. To find the scale factor, we can return to our two given corresponding lengths, 𝐴𝐵 and 𝑍𝑌. You might have already noticed that to go from 𝐴𝐵 to 𝑍𝑌, we would multiply the length by two. So the scale factor from 𝐴𝐵𝐶𝐷 to 𝑍𝑌𝑋𝐿 would be two.
If instead we wanted to go from the larger polygon to the smaller polygon, we’d really be dividing the length by two. But we must always give the scale factor in terms of multiplying, and dividing by two is the same as multiplying by one-half. Therefore, to find the length of line segment 𝐶𝐷, we would take the length of line segment 𝑋𝐿 and simply multiply it by one-half. This would give us the same value of 123.1 centimeters that we’ve previously worked out. So either method would allow us to find the length of line segment 𝐶𝐷.