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

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Given that π΄π·/π·πΆ = 3/7 and the area of β³π΄π΅πΆ = 484 cmΒ², find the area of β³π΄π·πΈ.

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### Video Transcript

Given that π΄π· over π·πΆ is equal to three over seven and the area of triangle π΄π΅πΆ equals 484 square centimeters, find the area of triangle π΄π·πΈ.

In this question, what weβre actually looking at is two similar triangles. And remember, two triangles are similar if their corresponding angles are congruent and their sides are proportional. So, the only difference is their size. Now to prove that two triangles are similar, we need to prove that at least two angles are the same. Because if we have two congruent angles, the third must also be congruent.

So, first of all, we can see here that angle π΄ is common to both triangles. Thatβs triangle π΄πΈπ· and triangle π΄π΅πΆ. And since the sides π·πΈ and πΆπ΅ are parallel, angles π΄πΈπ· and π΄π΅πΆ are corresponding angles. Theyβre congruent. And so we can say that triangles π΄π΅πΆ and π΄πΈπ· are similar since angle π΄ is a shared angle and angles π΄πΈπ· and π΄π΅πΆ are corresponding angles on parallel lines. We can notice too that angles π΄π·πΈ and π΄πΆπ΅ are also corresponding for the same reason.

So now we know that our triangles are similar, we can use this fact to solve the given problem of finding the area of the smaller triangle π΄π·πΈ. We do this using the scale factor method, which we can apply here because our two triangles are an enlargement of each other. And thatβs where mathematical enlargement can mean getting bigger or getting smaller.

Weβve been given the information that the ratio of side π΄π· to side π·πΆ is three to seven. But what we really want to do is to compare side π΄π· to side π΄πΆ. So, we want to find the ratio of π΄π· to π΄πΆ to give us the scale factor for the sides of the two triangles. And thatβs three over three plus seven, which is three over 10. But remember, this is the ratio of the sides π΄π· and π΄πΆ. Itβs not the actual lengths of the sides, just the number of parts when weβre looking at the ratio. So, for example, side π΄π· is not necessarily three units long; itβs just that there are three parts of π΄π· to every 10 parts of π΄πΆ.

So now that we have the ratio π΄π· to π΄πΆ is three-tenths, we can work out our scale factor for the areas. Weβve been given the area of the larger triangle π΄π΅πΆ; thatβs 484 square centimeters. And we need to work from this to find the area of the smaller triangle π΄π·πΈ. So, our enlargement means shrinking the area, and we need to find the appropriate scale factor for this. Now rearranging our equation by multiplying both sides by π΄πΆ, this gives us π΄π· equals three-tenths of π΄πΆ. So, we can say that the scale factor for the sides of our triangles, taking us from the bigger triangle π΄π΅πΆ to the smaller triangle π΄π·πΈ, is three-tenths.

And so, to help us solve our problem, since weβre actually looking for an area, not a length, we recall that the area scale factor is equal to the square of the linear scale factor. And we can see that our linear scale factor is three-tenths. Therefore, our area scale factor must be three-tenths squared, which is nine over 100.

π΄nd so, the area of the smaller triangle π΄π·πΈ is the area of the larger triangle π΄π΅πΆ, which is 484, times our area scale factor, nine over 100. And thatβs 43.56. So, we can say that the area of triangle π΄π·πΈ is equal to 43.56 centimeters squared.

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