Video Transcript
Given that 𝐴𝐵𝐶𝐷 is similar to
𝑍𝑌𝑋𝐿, find the measure of angle 𝑋𝐿𝑍 and the length of the line segment
𝐶𝐷.
We are told here that 𝐴𝐵𝐶𝐷 is
similar to 𝑍𝑌𝑋𝐿. And that means that their
corresponding angles are congruent and their corresponding sides are in
proportion. We are first asked to find the
measure of angle 𝑋𝐿𝑍. But we noticed that we don’t have
enough information about the angles in polygon 𝑍𝑌𝑋𝐿 to work out this angle
measure. However, as these polygons are
similar, we know that this angle measure of 𝐵𝐶𝐷 is corresponding to angle 𝑌𝑋𝐿,
and so the angle 𝑌𝑋𝐿 must have a measure of 85 degrees. We can then use the fact that the
internal angle measures in a quadrilateral add to 360 degrees to work out that the
measure of angle 𝑋𝐿𝑍 must be equal to 360 degrees subtract 85 degrees plus 109
degrees plus 105 degrees. This gives us 61 degrees.
Next, we need to find the length of
the line segment 𝐶𝐷. We note that the length of 𝐶𝐷
corresponds with the length 𝑋𝐿 in polygon 𝑍𝑌𝑋𝐿. The proportion of these sides will
be the same proportion as that between all other pairs of corresponding sides in the
polygon. When we are answering problems
involving similar polygons, we need to look for or be able to calculate another pair
of corresponding sides. Here we are given that 𝐴𝐵 is 75
centimeters and 𝑍𝑌 is 150 centimeters. We could therefore write that 𝐶𝐷
over 𝑋𝐿 is equal to 𝐴𝐵 over 𝑍𝑌.
Filling in the length measurements,
we have that 𝐶𝐷 over 246.2 is equal to 75 over 150. We can take out our common factor
of 75 from the numerator and denominator on the right-hand side. And then we can rearrange by
multiplying through by 246.2. The length of 𝐶𝐷 is therefore
123.1 centimeters. Alternatively, we could’ve found
the scale factor or ratio of enlargement between the two polygons. Using the side length of 𝑥𝑦 and
𝐴𝐵, we could’ve found that to go to the polygon 𝐴𝐵𝐶𝐷 from the polygon
𝑍𝑌𝑋𝐿, we multiply by one-half. Multiplying the length 246.2
centimeters by one-half would have given us the same value of 123.1 centimeters.
As we have now answered both parts
of this question, we can give the answer that the measure of angle 𝑋𝐿𝑍 is 61
degrees and the length of 𝐶𝐷 is 123.1 centimeters.
