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### Perimeter & Area of Similar Figures

- If two figures have a scale factor of a : b, then the perimeters of the two figures will also be a : b
- If two figures have a scale factor of a : b, then the area of the two figures will be a
^{2}: b^{2}

### Perimeter & Area of Similar Figures

^{2}:13

^{2}= 144:169

∆ABC is similar to ∆DEF, find the ratio of their areas.

^{2}:5

^{2}= 64:25

If the ratio of corresponding side lengths is a:b, then the ratio of areas is a:b.

- Ratio of perimeters: 13:3
- Ratio of areas: 169:9

Ratio of areas: 169:9

ABCD is similar to EFGH, the ratio of their areas is 49:25, find BC.

- BC:FG = √{49} :√{25}
- BC:FG = 7:5

Trapezoid ABCD is similar to trapezoid EFGH, the ratio of their perimeters is 7:4, find AB.

- AB:EF = 7:4
- EF = 8

Given the ratio of corresponding side lengths of two polygons, we can find the ratio of their perimeters and the ratio of their areas.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Perimeter & Area of Similar Figures

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Perimeter of Similar Figures 0:08
- Example: Scale Factor & Perimeter of Similar Figures
- Area of Similar Figures 2:44
- Example:Scale Factor & Area of Similar Figures
- Extra Example 1: Complete the Table 6:09
- Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures 8:56
- Extra Example 3: Find the Unknown Area 12:04
- Extra Example 4: Use the Given Area to Find AB 14:26

### Geometry Online Course

### Transcription: Perimeter & Area of Similar Figures

*Welcome back to Educator.com.*0000

*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 side*0151

*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 a ^{2}:b^{2}.*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 a ^{2}:b^{2}.*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 a ^{2}:b^{2}.*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 factor*0252

*of the areas between this one and this one is going to be 2 ^{2}:3^{2}, 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 a ^{2}:b^{2}.*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 a ^{2}:b^{2}; be careful not to multiply it by 2--you have to square it.*0424

*So, 4 ^{2} is 16; and 7^{2} 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 3 ^{2} to 2^{2}; that is 9:4.*0469

*And the third one: they give us the ratio of the areas, so since this is a ^{2} to b^{2}, 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 13 ^{2} is 169.*0497

*And 144...the square root of that is 12; 12 ^{2} = 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 3 ^{2}, which is 9, and 1^{2}, 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 areas*0621

*(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 a ^{2} to b^{2}; 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 a ^{2} to b^{2},*0911

*which is a ^{2} to...b is 8, so 8^{2}.*0929

*Now, that is the same thing as a ^{2}/8^{2}; so we are going to use this ratio and make it equal to these areas.*0938

*So, a ^{2} is the same thing as, here, 218, over 166; so the ratio of this area to that area is a^{2}: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) a ^{2} equals 218 times 8^{2} (is 64).*0975

*So, from here, you can just divide this 166; a ^{2} =...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 a ^{2} to b^{2}, 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

0 answers

Post by Ramez Hajelsawi on February 3, 2013

this helped me a lot!!!!!