Question Video: Finding the Area of Similar Triangles given the Ratio between Their Side Lengths | Nagwa Question Video: Finding the Area of Similar Triangles given the Ratio between Their Side Lengths | Nagwa

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

## Join Nagwa Classes

Given that ๐ด๐ท/๐ท๐ถ = 3/2 and the area of โณ๐ด๐ต๐ถ = 695 cmยฒ, find the area of trapezoid ๐ท๐ถ๐ต๐ธ.

05:50

### Video Transcript

Given that ๐ด๐ท over ๐ท๐ถ equals three over two and the area of triangle ๐ด๐ต๐ถ equals 695 centimeters squared, find the area of trapezoid ๐ท๐ถ๐ต๐ธ.

So in this problem, what weโre looking at are two similar triangles. Weโve got the triangle ๐ด๐ท๐ธ and a triangle ๐ด๐ต๐ถ. 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 ๐ด. And then what we also know is because we have two parallel lines, which are denoted here by these arrowheads, and thatโs the two parallel lines ๐ท๐ธ and ๐ถ๐ต, then in fact the angle ๐ด๐ธ๐ท is going to be equal to the angle ๐ด๐ต๐ถ because these are corresponding angles.

So therefore, we can say that triangle ๐ด๐ธ๐ท is similar to triangle ๐ด๐ต๐ถ. And weโve done that using the angle-angle proof. And thatโs because if we have two angles the same, then the third angle must be the same. And thatโ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 โ thatโs angle ๐ด๐ท๐ธ and angle ๐ด๐ถ๐ต โ because once again theyโ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 weโre going to use to solve this problem is scale factor. So weโre gonna find the scale factor of enlargement.

So before we do that, letโs take a look at the information we have. So we know that ๐ด๐ท over ๐ท๐ถ is equal to three over two. So therefore, what we can say is that ๐ด๐ท over ๐ด๐ถ is going to be equal to three over five. And thatโs because if we think about it, ๐ด๐ท is represented by our three and our ๐ท๐ถ is represented by our two. And what we must remember that this is not necessarily the length of ๐ด๐ท and ๐ท๐ถ. Itโs just what itโs represented by in our fraction or ratio that we have. Well, therefore, if three parts is ๐ด๐ท and two parts is ๐ท๐ถ, then ๐ด๐ถ must be three plus two, which is five parts. So therefore, we can say that ๐ด๐ท over ๐ด๐ถ is gonna be equal to three over five.

So then what we can do is multiply through by ๐ด๐ถ. When we do that, we get ๐ด๐ท is equal to three-fifths multiplied by ๐ด๐ถ. And then if we divide by three-fifths, what we would get is that five-thirds multiplied by ๐ด๐ท โ and thatโs because if you divide by three over five, itโs the same as multiplying by five over three โ is going to be equal to ๐ด๐ถ. So what we can say is that to get ๐ด๐ถ from ๐ด๐ท, and thatโ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, weโre not actually looking to find a length of one of the sides or a length of our trapezoid. What weโ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 ๐ด๐ท๐ธ โ so this is the area of the smaller triangle, so weโre going from the bigger triangle to the smaller triangle โ is going to be equal to the area of the bigger triangle ๐ด๐ต๐ถ, 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 weโre looking to find. In fact, what weโre looking to find is the area of trapezoid ๐ท๐ถ๐ต๐ธ, which Iโve shaded here in blue. And to find this, what weโre going to do is subtract the area of the smaller triangle ๐ด๐ท๐ธ away from the area of the larger triangle ๐ด๐ต๐ถ. So weโ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 ๐ท๐ถ๐ต๐ธ is going to be 444.8 centimeters squared.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions