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 congruent and similar solids.
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Whenever we have two solids that are either similar or congruent, there is a scale factor.
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A scale factor is just the ratio that compares the two solids; it is the ratio of the corresponding measures (it has to be corresponding).
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If we are going to use this side for the scale factor, then we have to use the corresponding side of the other solid.
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So then again, the scale factor is the ratio of the two similar solids.
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Here, the scale factor...since this is 2 and this is 4, we are going to say that it is 2:4; simplified, this is 1:2.
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The scale factor is 1:2, or we can say 1 to 2, like that; it is just the ratio between the two similar solids.
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For congruent solids, these have to be true: the corresponding angles are congruent; the corresponding edges are congruent
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(we have to have congruency between the two solids); the areas of the faces are congruent; and the volumes have to be congruent.
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And congruent solids have the same size and same shape.
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Remember that, for congruent solids, it is same size and same shape; for similar solids, it is going to be different sizes, but same shape.
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Remember: whenever we have something similar, it has to be the same exact shape, but then just a different size.
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Congruent solids will have the same shape and the same size.
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And the scale factor is going to be 1:1, because obviously, the corresponding sides and the corresponding parts are going to be the same.
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So, it is going to be a ratio of 1:1.
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Looking at similar solids, if the scale factor is a:b, then the ratio of the surface areas is going to be a²:b²,
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and the ratio of the volumes is going to be a³:b³.
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Let's say we have two solids, and the scale factor between the two is 2:3;
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then the ratio of the surface is going to be 2²:3², so it is going to be 4:9.
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If that is the scale factor (that is the ratio between the two solids), their surface areas are going to be 4:9.
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And then, the ratio of the volumes is going to be 2³:3³; 2 cubed is 8; 3 cubed is 27.
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That is going to be the ratio of their volumes.
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The first example is to determine whether each pair is similar, congruent, or neither.
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Looking at these two, this pair: here, this is a cube, because we know that all of the sides are going to be congruent.
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So, this is a cube; this is also a cube with all of the edges measuring 5.
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So, in this case, because they are the same shape, but just different sizes, this is similar.
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And we always know that all cubes are going to be similar, because cubes have the same shape.
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No matter how big or how small, all cubes are the same shape; they can just be different sizes.
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If they are the same shape, but same size, then they would be congruent; if all of these were also 8 inches, then they would be congruent.
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But since they just have the same shape and different sizes, they are just similar.
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And then, these two: let's see, here we have 24; that is diameter; from here to here is 26; we don't know the height.
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Remember: for these two to be congruent, they have to have congruent corresponding parts.
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To find the height here (because I don't know the height), I know the height here; this would be the height for this one,
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because it is just the cylinder that is turned sideways; so if we say that that is the height,
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then I need to find the height of this, so that I can compare.
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The diameter is 24 here; the radius is 12 there; so the radius here will also be 12, because the diameter is twice the length of the radius.
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To find the height here, I am going to use this triangle; and this is a right triangle, so then I can just use the Pythagorean theorem.
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It is going to be...if I name that h...h² + 24² is going to equal the hypotenuse (26) squared.
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So, 24 squared is 576; and then, 26 squared is 676; so, if I subtract them, I am going to get 100, which makes my height 10 centimeters.
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This is 10; this is also 10; so then, their heights are congruent; their radius is congruent.
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So, if I were to find the area of the base, then it is going to be π times 12 squared (the radius is 12, so it is 12 squared).
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Here, it is also going to be π times 12 squared; so the area of the bases will be the same.
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To find the volume, it is going to be the area of the base...that is π, r, squared, times the height.
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The same thing happens here: the radius is the same, and the height is the same, and we know that π is always the same.
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So then, their volumes are going to be exactly the same.
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Well, if we have two solids with the same exact volume, same shape, same size, same corresponding parts, we know that this has to be congruent.
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And again, because they are going to have the same volume, they are congruent.
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Determine if each statement is true or false: All spheres are similar.
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Spheres always have the same shape; no matter what, all spheres are the same shape.
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Now, sizes could vary; we could have a large sphere; we could have a small sphere.
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But they are always going to be similar, because they always have the same shape.
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So, any time two solids have the same shape, they are always going to be similar; so this is true.
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The next one: If two pyramids have square bases, then they must be similar.
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Well, even if they have a square base, yes, squares are always similar, because squares always have the same shape.
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A square is a square, whether it is large or small; they are always going to be similar.
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But for pyramids, we can have a tall pyramid, or we can have a short pyramid.
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So, it doesn't always mean that they are going to be similar--these do not have the same shape.
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So, even if their square bases are exactly the same (they are congruent), because we don't know the height, this is false.
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If two solids are congruent, then their volumes are equal.
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Well, let's say that we have exactly the same rectangular prism; it is congruent to this.
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If they are exactly the same, then isn't the space inside also going to be the same?
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So, all of this is going to be the same as all of this; so this is true.
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Congruent solids have congruent volumes, the same volume.
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For this example, we are going to find the scale factor and the ratio of the surface areas and the volume.
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The scale factor between this and this prism that are similar is going to be 4:6.
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We are going to use corresponding parts to determine the scale factor; it is going to be 4:6.
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We need to simplify this, and it is going to be 2:3; that is the scale factor between these two prisms.
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Then, to find the ratio of the surface area, for surface area, it is a²:b².
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And then, for volume, it is going to be a³:b³.
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For surface area, the ratio is going to be 2 squared to 3 squared, which is 4 to 9.
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Now, it doesn't mean that the surface area of this is 4 and the surface area of this is 9; it is just the ratio between this one and this one.
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So, when we find the surface area, if we were to find the surface area of both this prism and this prism,
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and then we simplify it, it is going to become 4:9.
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And then, for volume, it will be 2³ to...what is that one, 3?...3³, so it is going to be 8:27.
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And again, that does not mean that the volume of this is 8, and that the volume of this is 27.
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Let's actually find the volume, given two corresponding sides right here.
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This is similar, so the ratio between these two prisms is 3:2.
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And make sure that you keep the ratio the same; if you are going to keep it at 3:2, that means that you are listing out this one first.
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You are naming this first; so it is this one to that one.
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If you want to go the other way, that is fine; but then, you are going to have to make the scale factor 2:3, instead of 3:2.
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Always keep in mind which one this is: this number refers to the larger prism.
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Now, the volume of the smaller one, the second one, is given; it is 50 inches cubed.
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So, to find the ratio of the volumes, it is 3 cubed to 2 cubed; that is 27 to 8; that is the ratio of the volumes.
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This is the larger one, over the smaller one; that means that, if I want to find the actual volume,
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the volume of this one to the volume of this is going to become 27/8, simplified.
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Then, I just know that I can make a proportion: this ratio is going to equal the volume of that
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(because that one applies to the larger one), so let's say V for volume, over...what is the volume of this smaller one? 50.
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That is how I make my proportion, because the volume of the larger to the volume of the smaller, simplified, is going to become 27/8.
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Use the volume of the larger over the actual volume of the smaller prism.
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So then, here I am going to solve out this proportion; this becomes 8V (cross-products: 8 times V) equals 27 times 50.
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Using your calculator, 27 times 50 equals 1350; divide the 8; your volume is 168.75.
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And then here, our units are inches (and for volume, it has to be) cubed--units cubed.
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And that would be the volume of this larger prism.
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OK, so again, to make your proportion, we know that this ratio has to equal this ratio.
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They both are the ratios of their volumes; you have the volume of the larger prism to the volume of the smaller prism;
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that is going to become 27/8; so this is simplified, but then their volumes have to equal 27/8; that is the ratio of the prisms.
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You just make the two ratios equal to each other; set it equal.
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Make sure that you keep the larger prism as your numerator; so it has to be 27 over the smaller prism.
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And this is going to be V/50, the larger over the smaller.
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If you do it the other way, if you do the smaller over the larger, then you have to make sure that you flip this one, also: 50/V.
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Find cross-products, and then just solve it out.
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That is it for this lesson; thank you for watching Educator.com.