### Video Transcript

Given that π΄π΅πΆπ· is similar to ππππΏ, find the measure of angle ππΏπ and the length of line segment πΆπ·.

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 ππΏπ, which will be here on the larger polygon. The angle on the smaller polygon π΄π΅πΆπ· thatβs corresponding to ππΏπ will be this angle at πΆπ·π΄.

On some diagrams, itβs not easy to recognize which angles are corresponding, but we can always do it from the similarity statement if weβre given one. For example, in our similarity statement, angle ππΏπ will be corresponding with angle πΆπ·π΄. But here the problem is we donβt actually know the angle at πΆπ·π΄, so letβs see if we can find another pair of corresponding angles. We might notice that thereβs this angle of 85 degrees in the smaller polygon; itβs the angle π·πΆπ΅. And the corresponding angle in the larger polygon will be the angle πΏππ. So the angle πΏππ is also 85 degrees.

Now that we have three angles in this quadrilateral, we can find the measure of angle ππΏπ 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, weβll find the measure of angle ππΏπ. Calculating 360 degrees subtract 299 degrees gives us the angle of 61 degrees. And so thatβs the answer for the first part of the question to find the measure of angle ππΏπ.

We can now clear some space to find the length of the line segment πΆπ·. As weβre looking at the lengths of the sides in these two similar polygons, itβs 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 π΄π΅ is corresponding with the length of ππ. Usually, when weβre working with similar polygons, weβll 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 πΆπ·, so letβs look at the first method.

In this method, we would say that thereβs a proportion between the two corresponding sides of ππ and π΄π΅. And itβs going to be equal to the ratio or proportion between the line segment πΆπ· and its corresponding side. The side thatβs corresponding to πΆπ· will be this length of ππΏ. So weβre really saying that the proportion of ππ to π΄π΅ is the same as the proportion of ππΏ to πΆπ·.

Now all we need to do is fill in the values that weβre given for each of these lengths. So we have that 150 over 75 is equal to 246.2 over πΆπ·, which weβre 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 πΆπ· 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 πΆπ· is 123.1, and of course the units will be the length units of centimeters. And so thatβs the answer for the second part of this question.

But before we finish, letβs have a look at the second alternative method we couldβve used to find the length of line segment πΆπ·. 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, π΄π΅ and ππ. You might have already noticed that to go from π΄π΅ to ππ, we would multiply the length by two. So the scale factor from π΄π΅πΆπ· to ππππΏ would be two.

If instead we wanted to go from the larger polygon to the smaller polygon, weβd 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 πΆπ·, we would take the length of line segment ππΏ and simply multiply it by one-half. This would give us the same value of 123.1 centimeters that weβve previously worked out. So either method would allow us to find the length of line segment πΆπ·.