# Question Video: Finding the Side Length and Angle Measure in Similar Triangles Mathematics

If β³π΄π΅π· βΌ β³π΄πΆπ΅, find πβ π·π΅πΆ and the length of the line πΆπ· to the nearest tenth.

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### Video Transcript

If triangle π΄π΅π· is similar to triangle π΄πΆπ΅, find the measure of angle π·π΅πΆ and the length of πΆπ· to the nearest tenth.

In this question, weβre told that there are two triangles, π΄π΅π· and π΄πΆπ΅. And we could see this symbol here, which means that theyβre similar. We can recall that βsimilarβ means that corresponding pairs of angles are congruent or equal and corresponding sides are in proportion. If the corresponding sides were also congruent, then the two triangles would be congruent. But here, weβre told that theyβre similar.

Letβs start by getting a very clear idea about which two triangles weβre talking about. We have the first triangle, π΄π΅π·, which weβre told is similar to this larger triangle, π΄πΆπ΅. Sometimes it can be helpful to draw these triangles out separately so we can see them a bit more clearly.

Here we have the large triangle π΄π΅πΆ drawn. And itβs very important to keep the letters on our new diagram. We can also fill in the information about the angles and sides that we were given. When it comes to drawing the diagram for this second triangle, Iβm going to draw it in a different orientation. This is because itβs a little bit more helpful in helping us visualize the corresponding sides.

On triangle π΄π΅π·, the vertex π΄ is at the top of our diagram. We can see that the vertices π΄ and π΄ are similar or corresponding in both triangles. The angle at πΆ on triangle π΄πΆπ΅ is corresponding with the angle at π΅ on triangle π΄π΅π·. And finally, angle π΅ is corresponding with angle π· on triangle π΄π΅π·.

It can be particularly confusing in questions like this when we expect that the angle at π΅ is corresponding in both triangles. But itβs not in this question. We can fill in the information about the angle at π·π΄π΅, which is 77 degrees. And weβre also given the length of π΄π΅. Note that as these two triangles are similar with congruent angles, we could also fill it in this angle at π΄π΅π·. Thatβs 32 degrees.

So letβs look at the first part of this question to find the measure of angle π·π΅πΆ. We can see it here on our original diagram. But unfortunately, itβs not in triangle π΄π΅π·. And it would only form part of this triangle π΄π΅πΆ. But itβs not a whole angle that we can find from there. There are a few ways in which we could do this.

One method would involve finding this whole angle of π΄π΅πΆ, which occurs in the large triangle, and then subtracting this angle of π΄π΅π·. We can see that angle π΄π΅π· is 32 degrees. So letβs see if we can find this angle π΄π΅πΆ. We can use the fact that the angles in a triangle add up to 180 degrees to give us that angle π΄π΅πΆ must be equal to 180 degrees subtract 77 degrees subtract 32 degrees.

Carrying this calculation out, we find that angle π΄π΅πΆ is 71 degrees. So remembering that we want to find this pink angle π·π΅πΆ, we can do this with the larger angle π΄π΅πΆ and subtracting this angle at π΄π΅π·. The angle π΄π΅πΆ is the one weβve just calculated as 71 degrees. And angle π΄π΅π· is this one, which is 32 degrees. This gives us an answer of 39 degrees. And thatβs our answer for the first part of the question, to find the measure of angle π·π΅πΆ.

Another method that we couldβve used is to work out that this angle π΄π·π΅ is also 71 degrees, which we couldβve done using the angles in a triangle. Then, using the fact that the angles on a straight line add up to 180 degrees, we wouldβve worked out that this angle πΆπ·π΅ is 109 degrees. If we then considered this triangle π΅πΆπ·, and remembering that the angles add up to 180 degrees, we could find angle π·π΅πΆ by having 180 degrees subtract 109 degrees subtract 32 degrees, which wouldβve also given us 39 degrees, confirming our original answer.

Weβre now going to look at the second part of this question to find the length of πΆπ·. And Iβm going to delete some of the working if you want to pause the video to make any notes. If we look at the original diagram, we can see that this length πΆπ· doesnβt occur on our smaller triangle of π΄π΅π·. Itβs also only part of the line π΄πΆ on the larger triangle.

We can observe, however, that weβre given the length of π΄πΆ. And if we worked out this length of π΄π·, we could subtract it from the length π΄πΆ to find πΆπ·. Unfortunately, weβre not given this length of π΄π·. But we can use the fact that we have similar triangles in order to work it out.

We use the fact that corresponding sides in similar triangles are in proportion. Fortunately, however, we are given the lengths of two corresponding sides. We have π΄πΆ on triangle π΄πΆπ΅ and π΄π΅ on triangle π΄π΅π·. We could write this proportion as π΄π΅ over π΄πΆ. We have this length π΄π·, which we wish to calculate. The corresponding length on triangle π΄πΆπ΅ would be the length of π΄π΅.

As we know that corresponding sides have the same proportion, then we can put these proportions equal to each other. We can then fill in the lengths that weβre given. π΄π΅ is 19 centimeters, and π΄πΆ is 34 centimeters. We donβt know the length of π΄π·. Thatβs the one we want to find out. And the length of π΄π΅ is 19 centimeters.

We can solve for π΄π· by taking the cross product. 34 times π΄π· is 34π΄π·. And thatβs equal to 19 times 19. Evaluating the right-hand side gives us 361. We can then divide both sides of our equation by 34. And as weβre asked for the length of πΆπ·, which weβll do shortly, to the nearest tenth, we can assume that we could use a calculator here.

And so we find that π΄π· equals 10.61747 and so on centimeters. Weβre not going to write this value just yet as weβll use it in the next calculation. If we have a look at our diagram then, we can see that now we have found this value of π΄π·. Then we can find πΆπ· by subtracting our value 10.617 et cetera from 34. Evaluating this gives us a value of 23.38235 and so on centimeters.

Now we can write our value. Weβre rounding to the nearest tenth. So we check our second decimal digit to see if itβs five or more. And as it is, then our answer rounds up to 23.4 centimeters. So now we have the answer for both parts of the question. The measure of angle π·π΅πΆ is 39 degrees. And the length of πΆπ· was 23.4 centimeters to the nearest tenth.

This question was quite tricky in terms of working out which angles were corresponding. For example, we saw that this angle π΅ in our first triangle did not correspond with the angle π΅ in our other triangle. It is worth spending a little bit longer just to make sure that we get the diagrams right so that they can really help us.