Geometry Perimeter & Area of Similar Figures

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For the next lesson, we are going to go over perimeter and area of similar figures.0002

If you remember from similar polygons, they have a ratio, a scale factor.0010

A scale factor is the same thing as ratio of the corresponding parts, a:b.0019

Now, if the scale factor is a:b...so let's say, for example, that this is 2, and the corresponding side for this triangle is 3--0027

again, they are similar...then the ratio, the scale factor between them, is going to be 2:3.0040

Well, then the perimeter of this first one: if the scale factor is 2:3,0049

then the scale factor of the perimeter of the first one to the second one is also going to be...0055

the ratio, not the actual perimeter, but the ratio of the perimeters is going to be 2:3; it is going to be the same.0066

For example, if the perimeter of this is 5, well, we can turn this into a fraction; so 2:3 is going to be 2/3, like that;0075

since the ratio of the corresponding parts is the same as the ratio of the perimeters,0095

I can just make it equal to 5/P, to find the perimeter of this.0106

That way, I can just cross-multiply here; if we just make this equal to P, and leave that as P, 2 times P is 2P; that equals 3 times 15.0119

I can divide the 2, and then the perimeter is going to be 15/2, which is 7.5, 7 and 1/2.0136

So, if the perimeter is 5 here, then the perimeter of this has to be 7.5.0146

Again, the ratio is going to be the same; the scale factor of the corresponding parts of this side to this side0151

is going to be the same exact scale factor of the perimeters.0157

Now, for area, it is a little bit different: if the scale factor of this triangle to this triangle is a:b, then the area of the two figures--0166

the scale factor of the area--is going to be a²:b².0180

If this is a:b, if they are similar, then of course, the scale factor, the ratio, is going to be a:b.0188

Well, then, for this, if the scale factor of the area to the area...the area for the first one of triangle 1, let's say,0197

to the area of triangle 2, is going to be a²:b².0216

Now, that is not actually saying that that is going to be the actual area; just because you have a:b,0226

if you square those numbers, that doesn't mean that that is going to be the actual area for the triangles.0234

It is saying that the ratio between the two areas is going to be a²:b².0239

Let's say that a is this side right here; it is 2, and this side is 3.0247

So, the scale factor between these two triangles is going to be 2:3; that means that the scale factor0252

of the areas between this one and this one is going to be 2²:3², so it is going to be 4:9.0262

Now, it does not mean that the area of this triangle is going to be 4; it is saying that the ratio of the area from this one to this one is going to be 4:9.0276

So, if the area of this is 16 units squared, then how can I find the area of this one?0286

Let's say that the area of this is what we are looking for.0302

Since I know that the ratio of the area from this one to this one is going to be 4:9, I can just create a proportion.0308

So, 4:9 is going to equal 16 (because this top number is representing this triangle; this is representing this triangle) over x.0315

We are going to label that x; then you can cross-multiply.0329

Or, since we know that 4 is a factor of 16, to get from 4 to 16, I can just multiply this by 4, which means that to get x, I can just multiply this by 4.0335

So, this will be 36; so my area here is going to be 36 units squared.0353

Now, let's just go over some examples: The ratio of the corresponding side lengths is 4:7.0369

If this one is a:b, the ratio of the perimeter is also going to be a:b; the ratio of the areas, then, is going to be a²:b².0380

So, back to the first one: 4:7; the ratio of the perimeters is going to be 4:7.0399

Now again, that does not mean that the perimeter is going to be 4 units, and the perimeter of the second one is going to be 7 units.0407

It just means that when you simplify it, it is going to have a ratio of 4:7.0418

And then, the ratio of the areas is going to be a²:b²; be careful not to multiply it by 2--you have to square it.0424

So, 4² is 16; and 7² is 49; so this is going to be the ratio of the areas.0433

Again, it does not mean that these are going to be the areas; it just means that, when the areas are simplified, it is going to have the scale factor of 1009.0444

OK, and then here, for the second one, they give us the ratio of the perimeters.0456

This is a:b; this is also a:b; so this is going to stay at 3:2.0462

Then, the ratio of the areas is going to be 3² to 2²; that is 9:4.0469

And the third one: they give us the ratio of the areas, so since this is a² to b², I have to take the square root,0480

do the opposite of squaring (that is taking the square root of each of these).0490

If I take the square root of this, I am going to get 13, because 13² is 169.0497

And 144...the square root of that is 12; 12² = 144.0504

Then, the ratio of the corresponding side lengths is also going to be 0792.0511

And then, for the last one, here is the ratio of the perimeters; it stays 9:10.0520

And then, the ratio of the areas...square each of those...is going to be 81:100.0526

Here, they ask for the ratios of the perimeter and the area of the similar figures.0538

Here, we have a rectangle; so if this is 6, I know that this also has to be 6.0546

And here, also, if this is 2, then this also has to be 2.0554

And I know that this side with this side is corresponding; so the ratio is going to be 6:2.0561

But then, I have to simplify: that is going to be 3:1--here is the ratio of the corresponding parts.0572

For the perimeter, the ratio is also going to be 3:1.0583

And then, the area is going to be 3², which is 9, and 1², which stays 1.0599

Now, all they wanted is the ratio of the perimeter and the ratio of the area.0614

But here, this area is given; it is 24 inches squared; so what you can do...since we know the ratio of the areas0621

(this is 9:1), the actual area for this one is given; so we can use that to look for and find this area here.0631

So again, 9:1 is going to be 9/1; change that so that, that way, we can make equivalent ratios, and that will be a proportion.0646

The area of this one is 24, and then the area of this is going to be x.0659

So here, we can cross-multiply; this is going to be 9x = 24; if we divide the 9 from both sides, then I am going to get...0668

and here, you can just simplify; this is going to be 8/3.0685

You can change this to a mixed number if you like; so then, this is going to be 2 and 2/3.0693

The area of this is going to be 2 and 2/3 inches squared.0707

The next example: Find the unknown area.0726

We have the area of this, but we don't have the area of this, so this is the unknown area.0729

Here, this is corresponding with this; so the ratio between these two figures is going to be 6:8, which simplifies to 3:4.0736

So, the ratio of this to this is 3:4; now, the ratio of the areas (I am going to write the areas separately from that)...0751

this is a:b; the ratio of this area to this area is a² to b²; that is 9:16--that is the ratio of the areas.0764

The actual area is 54 here, and I need to find this right here; so this is going to be, let's say, x.0783

I am going to make this into a proportion: 9/16, or 9:16, equals 54:x.0794

You can cross-multiply; you can also...if this is a factor of this number, then to get from 9 to 54, you multiply by 6.0806

So, to get from 16 to x, you can just multiply by 6; and let's see, 16 (I have a calculator here) times 6 equals 96.0819

So x, this measure right here, is going to be 96; the area is 96 meters squared.0829

Again, we found the ratio of the areas; it is going to be 9:16, and we just use that to create a proportion.0844

So, 54:96 is going to be the same as 9:16.0855

And the last example: Use the given area to find AB.0868

So, this is what we are looking for, here: the area is given here; the area is given here.0874

This is also given; the corresponding side is given.0881

Let's label this as a and this as b; a:b would be the scale factor between the two figures.0885

We don't know a, but we know b; b is 8, so it is going to be a:8.0900

And a is what we are looking for, because that is AB.0906

Now, I know that, for the areas, it is going to be the scale factor squared; so it is a² to b²,0911

which is a² to...b is 8, so 8².0929

Now, that is the same thing as a²/8²; so we are going to use this ratio and make it equal to these areas.0938

So, a² is the same thing as, here, 218, over 166; so the ratio of this area to that area is a²:64.0954

And you are just going to use this proportion to solve.0971

It is going to be 166 (and I am just cross-multiplying) a² equals 218 times 8² (is 64).0975

So, from here, you can just divide this 166; a² =...and you can just use your calculator...218 times 64...0991

divide that number by 166, and I get 84.05.1011

And then, since we are solving for a, we need to take the square root of that;1023

so on your calculator, you can just take the square root of it; and I get 9.17.1029

So, this right here is going to be 9.17 centimeters.1039

Again, all I did was to label this a and b; the scale factor is a:8; to find the scale factor of the areas,1048

you are going to do a² to b², which is equal to 210966.1058

And then, solve it for the a; that is what we labeled as our AB, and that is centimeters.1070

Let me just rename this, since it is asking for AB; I'll say AB is 9.17 centimeters.1081

That is it for this lesson; thank you for watching Educator.com.1094