If triangle 𝐴𝐵𝐷 is similar to triangle 𝐴𝐶𝐵, find the measure of angle 𝐷𝐵𝐶 and the length of 𝐶𝐷 to the nearest tenth.
In this question, we’re told that there are two triangles, 𝐴𝐵𝐷 and 𝐴𝐶𝐵. And we could see this symbol here, which means that they’re similar. We can recall that “similar” 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, we’re told that they’re similar.
Let’s start by getting a very clear idea about which two triangles we’re talking about. We have the first triangle, 𝐴𝐵𝐷, which we’re told is similar to this larger triangle, 𝐴𝐶𝐵. 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 𝐴𝐵𝐶 drawn. And it’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, I’m going to draw it in a different orientation. This is because it’s a little bit more helpful in helping us visualize the corresponding sides.
On triangle 𝐴𝐵𝐷, the vertex 𝐴 is at the top of our diagram. We can see that the vertices 𝐴 and 𝐴 are similar or corresponding in both triangles. The angle at 𝐶 on triangle 𝐴𝐶𝐵 is corresponding with the angle at 𝐵 on triangle 𝐴𝐵𝐷. And finally, angle 𝐵 is corresponding with angle 𝐷 on triangle 𝐴𝐵𝐷.
It can be particularly confusing in questions like this when we expect that the angle at 𝐵 is corresponding in both triangles. But it’s not in this question. We can fill in the information about the angle at 𝐷𝐴𝐵, which is 77 degrees. And we’re also given the length of 𝐴𝐵. Note that as these two triangles are similar with congruent angles, we could also fill it in this angle at 𝐴𝐵𝐷. That’s 32 degrees.
So let’s look at the first part of this question to find the measure of angle 𝐷𝐵𝐶. We can see it here on our original diagram. But unfortunately, it’s not in triangle 𝐴𝐵𝐷. And it would only form part of this triangle 𝐴𝐵𝐶. But it’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 𝐴𝐵𝐶, which occurs in the large triangle, and then subtracting this angle of 𝐴𝐵𝐷. We can see that angle 𝐴𝐵𝐷 is 32 degrees. So let’s see if we can find this angle 𝐴𝐵𝐶. We can use the fact that the angles in a triangle add up to 180 degrees to give us that angle 𝐴𝐵𝐶 must be equal to 180 degrees subtract 77 degrees subtract 32 degrees.
Carrying this calculation out, we find that angle 𝐴𝐵𝐶 is 71 degrees. So remembering that we want to find this pink angle 𝐷𝐵𝐶, we can do this with the larger angle 𝐴𝐵𝐶 and subtracting this angle at 𝐴𝐵𝐷. The angle 𝐴𝐵𝐶 is the one we’ve just calculated as 71 degrees. And angle 𝐴𝐵𝐷 is this one, which is 32 degrees. This gives us an answer of 39 degrees. And that’s our answer for the first part of the question, to find the measure of angle 𝐷𝐵𝐶.
Another method that we could’ve used is to work out that this angle 𝐴𝐷𝐵 is also 71 degrees, which we could’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 would’ve worked out that this angle 𝐶𝐷𝐵 is 109 degrees. If we then considered this triangle 𝐵𝐶𝐷, and remembering that the angles add up to 180 degrees, we could find angle 𝐷𝐵𝐶 by having 180 degrees subtract 109 degrees subtract 32 degrees, which would’ve also given us 39 degrees, confirming our original answer.
We’re now going to look at the second part of this question to find the length of 𝐶𝐷. And I’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 𝐶𝐷 doesn’t occur on our smaller triangle of 𝐴𝐵𝐷. It’s also only part of the line 𝐴𝐶 on the larger triangle.
We can observe, however, that we’re given the length of 𝐴𝐶. And if we worked out this length of 𝐴𝐷, we could subtract it from the length 𝐴𝐶 to find 𝐶𝐷. Unfortunately, we’re not given this length of 𝐴𝐷. 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 𝐴𝐶 on triangle 𝐴𝐶𝐵 and 𝐴𝐵 on triangle 𝐴𝐵𝐷. We could write this proportion as 𝐴𝐵 over 𝐴𝐶. We have this length 𝐴𝐷, which we wish to calculate. The corresponding length on triangle 𝐴𝐶𝐵 would be the length of 𝐴𝐵.
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 we’re given. 𝐴𝐵 is 19 centimeters, and 𝐴𝐶 is 34 centimeters. We don’t know the length of 𝐴𝐷. That’s the one we want to find out. And the length of 𝐴𝐵 is 19 centimeters.
We can solve for 𝐴𝐷 by taking the cross product. 34 times 𝐴𝐷 is 34𝐴𝐷. And that’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 we’re asked for the length of 𝐶𝐷, which we’ll do shortly, to the nearest tenth, we can assume that we could use a calculator here.
And so we find that 𝐴𝐷 equals 10.61747 and so on centimeters. We’re not going to write this value just yet as we’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 𝐴𝐷. Then we can find 𝐶𝐷 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. We’re rounding to the nearest tenth. So we check our second decimal digit to see if it’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 𝐷𝐵𝐶 is 39 degrees. And the length of 𝐶𝐷 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 𝐵 in our first triangle did not correspond with the angle 𝐵 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.