Question Video: Finding the Area of Similar Triangles given the Ratio between Their Side Lengths Mathematics • First Year of Secondary School

Video Transcript

Given that 𝐴𝐷 over 𝐷𝐶 equals three over two and the area of triangle 𝐴𝐵𝐶 equals 695 centimeters squared, find the area of trapezoid 𝐷𝐶𝐵𝐸.

So in this problem, what we’re looking at are two similar triangles. We’ve got the triangle 𝐴𝐷𝐸 and a triangle 𝐴𝐵𝐶. And when we are dealing with similar triangles, then what this means is that we have one triangle which is an enlargement or dilation of the other. So they are in fact in proportion and have all the corresponding angles equal.

And we can prove that in this problem because first of all we have one shared angle at 𝐴. And then what we also know is because we have two parallel lines, which are denoted here by these arrowheads, and that’s the two parallel lines 𝐷𝐸 and 𝐶𝐵, then in fact the angle 𝐴𝐸𝐷 is going to be equal to the angle 𝐴𝐵𝐶 because these are corresponding angles.

So therefore, we can say that triangle 𝐴𝐸𝐷 is similar to triangle 𝐴𝐵𝐶. And we’ve done that using the angle-angle proof. And that’s because if we have two angles the same, then the third angle must be the same. And that’s because all the angles in a triangle add up to 180. And in fact in our problem, we know that the two angles are going to be the same — that’s angle 𝐴𝐷𝐸 and angle 𝐴𝐶𝐵 — because once again they’re corresponding angles.

Okay, great. So we now know the properties between our triangles, and that is that they are similar. Well, as the two triangles are similar to each other, we know that one is going to be an enlargement of the other. So the method we’re going to use to solve this problem is scale factor. So we’re gonna find the scale factor of enlargement.

So before we do that, let’s take a look at the information we have. So we know that 𝐴𝐷 over 𝐷𝐶 is equal to three over two. So therefore, what we can say is that 𝐴𝐷 over 𝐴𝐶 is going to be equal to three over five. And that’s because if we think about it, 𝐴𝐷 is represented by our three and our 𝐷𝐶 is represented by our two. And what we must remember that this is not necessarily the length of 𝐴𝐷 and 𝐷𝐶. It’s just what it’s represented by in our fraction or ratio that we have. Well, therefore, if three parts is 𝐴𝐷 and two parts is 𝐷𝐶, then 𝐴𝐶 must be three plus two, which is five parts. So therefore, we can say that 𝐴𝐷 over 𝐴𝐶 is gonna be equal to three over five.

So then what we can do is multiply through by 𝐴𝐶. When we do that, we get 𝐴𝐷 is equal to three-fifths multiplied by 𝐴𝐶. And then if we divide by three-fifths, what we would get is that five-thirds multiplied by 𝐴𝐷 — and that’s because if you divide by three over five, it’s the same as multiplying by five over three — is going to be equal to 𝐴𝐶. So what we can say is that to get 𝐴𝐶 from 𝐴𝐷, and that’s a length on the small triangle to a length on the big triangle, we multiply by five over three, or five-thirds. So therefore, we can say that the scale factor of enlargement or dilation is going to be five over three, or five-thirds.

Now, in this problem, we’re not actually looking to find a length of one of the sides or a length of our trapezoid. What we’re looking to deal with is area. So therefore, what we need to do is actually find the area scale factor. And what we know about the area scale factor is that the area scale factor is equal to the linear scale factor squared. So therefore, our area scale factor is gonna be equal to five-thirds squared, which is equal to 25 over nine.

So therefore, what we can say is that the area of our triangle 𝐴𝐷𝐸 — so this is the area of the smaller triangle, so we’re going from the bigger triangle to the smaller triangle — is going to be equal to the area of the bigger triangle 𝐴𝐵𝐶, which is 695, divided by our scale factor, our area scale factor in fact. And that is gonna be 25 over nine, which is gonna give us 250.2, and that would be centimeters squared.

Okay, have we solved the problem? Well no, this is not in fact what we’re looking to find. In fact, what we’re looking to find is the area of trapezoid 𝐷𝐶𝐵𝐸, which I’ve shaded here in blue. And to find this, what we’re going to do is subtract the area of the smaller triangle 𝐴𝐷𝐸 away from the area of the larger triangle 𝐴𝐵𝐶. So we’re going to have 695 minus 250.2, which is going to be equal to 444.8. So therefore, we can say that the area of trapezoid 𝐷𝐶𝐵𝐸 is going to be 444.8 centimeters squared.