Lesson video

Hi.

I'm Mr. Chan.

And in this lesson, we're going to learn about finding missing links in similar shapes, which have sides overlapping.

Here's an example where we've got similar shapes that have overlapping sides.

So we've got triangle ABC and BDE.

So, let's see if we can visualise those triangles there.

In terms of the triangle ABC, we can see ABC there in red, and BDE would be the smaller triangle there.

So we can see that there are two triangles in the shape.

So we've got to say why they're similar.

So in order to find similar shapes, we've got to decide whether all pairs of corresponding angles are the same.

So, if we look at angle B, now that angle there is the same angle for both triangles, ABC and BDE.

So that's like a common angle if you like.

And if we look at angle BDE and angle BAC, those two angles would be the same because of the parallel base sides those triangles have.

What we know about angles and parallel lines, those two angles would be considered corresponding angles.

And what we know about corresponding angles is that they're equal.

Similarly, the angles BED and angles BCA are also corresponding because of the parallel lines on the base of those triangles.

So, they're equal.

And what we can now say is that the angles in triangle ABC and the angles in BDE all match up with each other.

And because the three pairs of equal angles are the same, what we can say is those two triangles are similar.

Here's a question for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here's the answer.

So the reason why these two triangles are similar is because the three pairs of angles are all equal to each other, and one of the similar triangles is an enlargement of the other one.

That's the reason why we can say they're similar.

Let's have a look at an example where we're working out a missing side length where the shapes are overlapping now.

So in this example, I need to work out side length AC.

What I like to do is draw the two similar shapes separately.

So here's my triangle, ABC, and my other triangle would be DBE.

Those two shapes are similar.

We can see that from the matching pairs of equal angles.

Now, what I also like to do is just make sure I put in the side lengths that I do know.

So in the triangle BDE, I know that the side lengths are eight centimetres and seven centimetres.

And also, fill in in the information for the larger triangle.

Length AB would be made up of those two eight centimetre sides.

So, I'm going to put 16 centimetres for side length AB.

Now, in order to figure out what side length AC is, I need to figure out what my scale factor for the enlargement would be.

Well, I can see that in the smaller triangle length BD corresponds with length AB, so I need to multiply eight by two to get 16.

So my scale factor is two.

So that means in order to find AC I multiply length DE by two.

So that's seven centimetres multiplied by two.

That gives me 14 centimetres.

So, I've worked out the length of AC.

That would be 14 centimetres.

Here's a question for you to try.

Remember that drawing the similar triangles separately might help you with this question.

Pause the video to complete the task.

Resume the video once you're finished.

Here's the answer for question two.

If you didn't get the correct answer, it might be a good idea to draw the two similar triangles separately.

And that will help you find that the scale factor is two in order to find that DE equals 16 centimetres.

Here's another question you can try.

Pause the video to complete the task.

Resume the video once you're finished.

Here's the answer.

So this question we have to work out the length of side EG.

And in order to do that, we have to work out the length of the long side DG first, then subtract the length of DE in order to get the final answer.

Here's another example where I've got to find a missing side length and we've got side lengths that overlap.

So in this example, I've got to work out the length of CD.

Now, I can see that there are two separate triangles there, but it's quite difficult to see how they're similar.

So let's look at finding the missing angles in those triangles.

I know that angles in a triangle add up to 180 degrees, so I can find those two missing angles.

They are 27 degrees and 63 degrees.

So I've got my missing angles.

Now let's draw the two triangles separately, just like every other example I've done with you.

I'm drawing triangle ABD there, and also triangle BDC.

Now, it still doesn't look very clear as to why these two triangles are similar.

However, if we imagine rotating the larger triangle anticlockwise, 90 degrees like this, we can quite clearly see that the pairs of angles now match up and are equal to each other.

The 63 degrees is in the same position as a 63 degrees, as is the 27 degrees, as is the right angle, 90 degrees.

So, that helps me find the scale factor, which I can see is two because the 3.

5 centimetre side is enlarged by a multiple times two to get seven centimetres.

So that's my scale factor.

So that tells me that the corresponding side length, the height of the triangle seven centimetres multiplied by two, gets me CD to be 14 centimetres.

So my final answer, length CD equals 14 centimetres.

Here's a question for you to try.

Remember that drawing the two similar triangles separately will help you answer these types of questions.

Pause the video to have a go.

Resume the video once you're finished.

Here's the answer.

So the example covered something very similar to this question.

So remember to find the missing angles in the triangles, draw the two similar triangles separately, and then if they aren't oriented properly in the orientation that you can see them as similar, then rotate one of the triangles to match the pairs of angles up.

That way, you can find the scale factor and then find the missing length.

That's all for this lesson.

Thanks for watching.