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

Given that 𝐴𝐷/𝐷𝐶 = 3/7 and the area of △𝐴𝐵𝐶 = 484 cm², find the area of △𝐴𝐷𝐸.

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Video Transcript

Given that 𝐴𝐷 over 𝐷𝐶 is equal to three over seven and the area of triangle 𝐴𝐵𝐶 equals 484 square centimeters, find the area of triangle 𝐴𝐷𝐸.

In this question, what we’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 𝐴 is common to both triangles. That’s triangle 𝐴𝐸𝐷 and triangle 𝐴𝐵𝐶. And since the sides 𝐷𝐸 and 𝐶𝐵 are parallel, angles 𝐴𝐸𝐷 and 𝐴𝐵𝐶 are corresponding angles. They’re congruent. And so we can say that triangles 𝐴𝐵𝐶 and 𝐴𝐸𝐷 are similar since angle 𝐴 is a shared angle and angles 𝐴𝐸𝐷 and 𝐴𝐵𝐶 are corresponding angles on parallel lines. We can notice too that angles 𝐴𝐷𝐸 and 𝐴𝐶𝐵 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 𝐴𝐷𝐸. We do this using the scale factor method, which we can apply here because our two triangles are an enlargement of each other. And that’s where mathematical enlargement can mean getting bigger or getting smaller.

We’ve been given the information that the ratio of side 𝐴𝐷 to side 𝐷𝐶 is three to seven. But what we really want to do is to compare side 𝐴𝐷 to side 𝐴𝐶. So, we want to find the ratio of 𝐴𝐷 to 𝐴𝐶 to give us the scale factor for the sides of the two triangles. And that’s three over three plus seven, which is three over 10. But remember, this is the ratio of the sides 𝐴𝐷 and 𝐴𝐶. It’s not the actual lengths of the sides, just the number of parts when we’re looking at the ratio. So, for example, side 𝐴𝐷 is not necessarily three units long; it’s just that there are three parts of 𝐴𝐷 to every 10 parts of 𝐴𝐶.

So now that we have the ratio 𝐴𝐷 to 𝐴𝐶 is three-tenths, we can work out our scale factor for the areas. We’ve been given the area of the larger triangle 𝐴𝐵𝐶; that’s 484 square centimeters. And we need to work from this to find the area of the smaller triangle 𝐴𝐷𝐸. 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 𝐴𝐶, this gives us 𝐴𝐷 equals three-tenths of 𝐴𝐶. So, we can say that the scale factor for the sides of our triangles, taking us from the bigger triangle 𝐴𝐵𝐶 to the smaller triangle 𝐴𝐷𝐸, is three-tenths.

And so, to help us solve our problem, since we’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.

𝐴nd so, the area of the smaller triangle 𝐴𝐷𝐸 is the area of the larger triangle 𝐴𝐵𝐶, which is 484, times our area scale factor, nine over 100. And that’s 43.56. So, we can say that the area of triangle 𝐴𝐷𝐸 is equal to 43.56 centimeters squared.

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