WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over perimeter and area of similar figures.
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If you remember from similar polygons, they have a ratio, a scale factor.
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A scale factor is the same thing as ratio of the corresponding parts, a:b.
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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--
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again, they are similar...then the ratio, the scale factor between them, is going to be 2:3.
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Well, then the perimeter of this first one: if the scale factor is 2:3,
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then the scale factor of the perimeter of the first one to the second one is also going to be...
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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.
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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;
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since the ratio of the corresponding parts is the same as the ratio of the perimeters,
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I can just make it equal to 5/P, to find the perimeter of this.
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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.
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I can divide the 2, and then the perimeter is going to be 15/2, which is 7.5, 7 and 1/2.
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So, if the perimeter is 5 here, then the perimeter of this has to be 7.5.
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Again, the ratio is going to be the same; the scale factor of the corresponding parts of this side to this side
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is going to be the same exact scale factor of the perimeters.
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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--
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the scale factor of the area--is going to be a²:b².
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If this is a:b, if they are similar, then of course, the scale factor, the ratio, is going to be a:b.
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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,
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to the area of triangle 2, is going to be a²:b².
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Now, that is not actually saying that that is going to be the actual area; just because you have a:b,
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if you square those numbers, that doesn't mean that that is going to be the actual area for the triangles.
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It is saying that the ratio between the two areas is going to be a²:b².
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Let's say that a is this side right here; it is 2, and this side is 3.
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So, the scale factor between these two triangles is going to be 2:3; that means that the scale factor
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of the areas between this one and this one is going to be 2²:3², so it is going to be 4:9.
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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.
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So, if the area of this is 16 units squared, then how can I find the area of this one?
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Let's say that the area of this is what we are looking for.
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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.
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So, 4:9 is going to equal 16 (because this top number is representing this triangle; this is representing this triangle) over x.
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We are going to label that x; then you can cross-multiply.
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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.
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So, this will be 36; so my area here is going to be 36 units squared.
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Now, let's just go over some examples: The ratio of the corresponding side lengths is 4:7.
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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.
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So, back to the first one: 4:7; the ratio of the perimeters is going to be 4:7.
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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.
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It just means that when you simplify it, it is going to have a ratio of 4:7.
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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.
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So, 4² is 16; and 7² is 49; so this is going to be the ratio of the areas.
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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.
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OK, and then here, for the second one, they give us the ratio of the perimeters.
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This is a:b; this is also a:b; so this is going to stay at 3:2.
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Then, the ratio of the areas is going to be 3² to 2²; that is 9:4.
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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,
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do the opposite of squaring (that is taking the square root of each of these).
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If I take the square root of this, I am going to get 13, because 13² is 169.
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And 144...the square root of that is 12; 12² = 144.
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Then, the ratio of the corresponding side lengths is also going to be 0792.
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And then, for the last one, here is the ratio of the perimeters; it stays 9:10.
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And then, the ratio of the areas...square each of those...is going to be 81:100.
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Here, they ask for the ratios of the perimeter and the area of the similar figures.
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Here, we have a rectangle; so if this is 6, I know that this also has to be 6.
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And here, also, if this is 2, then this also has to be 2.
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And I know that this side with this side is corresponding; so the ratio is going to be 6:2.
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But then, I have to simplify: that is going to be 3:1--here is the ratio of the corresponding parts.
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For the perimeter, the ratio is also going to be 3:1.
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And then, the area is going to be 3², which is 9, and 1², which stays 1.
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Now, all they wanted is the ratio of the perimeter and the ratio of the area.
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But here, this area is given; it is 24 inches squared; so what you can do...since we know the ratio of the areas
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(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.
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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.
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The area of this one is 24, and then the area of this is going to be x.
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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...
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and here, you can just simplify; this is going to be 8/3.
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You can change this to a mixed number if you like; so then, this is going to be 2 and 2/3.
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The area of this is going to be 2 and 2/3 inches squared.
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The next example: Find the unknown area.
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We have the area of this, but we don't have the area of this, so this is the unknown area.
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Here, this is corresponding with this; so the ratio between these two figures is going to be 6:8, which simplifies to 3:4.
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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)...
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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.
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The actual area is 54 here, and I need to find this right here; so this is going to be, let's say, x.
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I am going to make this into a proportion: 9/16, or 9:16, equals 54:x.
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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.
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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.
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So x, this measure right here, is going to be 96; the area is 96 meters squared.
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Again, we found the ratio of the areas; it is going to be 9:16, and we just use that to create a proportion.
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So, 54:96 is going to be the same as 9:16.
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And the last example: Use the given area to find AB.
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So, this is what we are looking for, here: the area is given here; the area is given here.
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This is also given; the corresponding side is given.
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Let's label this as a and this as b; a:b would be the scale factor between the two figures.
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We don't know a, but we know b; b is 8, so it is going to be a:8.
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And a is what we are looking for, because that is AB.
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Now, I know that, for the areas, it is going to be the scale factor squared; so it is a² to b²,
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which is a² to...b is 8, so 8².
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Now, that is the same thing as a²/8²; so we are going to use this ratio and make it equal to these areas.
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So, a² is the same thing as, here, 218, over 166; so the ratio of this area to that area is a²:64.
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And you are just going to use this proportion to solve.
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It is going to be 166 (and I am just cross-multiplying) a² equals 218 times 8² (is 64).
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So, from here, you can just divide this 166; a² =...and you can just use your calculator...218 times 64...
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divide that number by 166, and I get 84.05.
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And then, since we are solving for a, we need to take the square root of that;
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so on your calculator, you can just take the square root of it; and I get 9.17.
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So, this right here is going to be 9.17 centimeters.
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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,
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you are going to do a² to b², which is equal to 210966.
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And then, solve it for the a; that is what we labeled as our AB, and that is centimeters.
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Let me just rename this, since it is asking for AB; I'll say AB is 9.17 centimeters.
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That is it for this lesson; thank you for watching Educator.com.