WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this next lesson, we are going to go over similar polygons.
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Now, polygons, we know, are shapes that have three or more sides.
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They have to be enclosed, meaning that there can't be any open gaps; and remember that the sides can't overlap.
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And the sides do have to be straight.
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So then, we know that quadrilaterals are polygons; triangles, pentagons...all of those are polygons.
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Now, we are going to talk about similarity; we know that congruent polygons are polygons that are exactly the same--
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the same size and same shape; now, similar polygons are a little bit different.
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They have the same shape, but they are not the same size.
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Two polygons are similar if and only if their corresponding angles are congruent, and the measures of their corresponding sides are proportional.
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In order for two polygons to be similar, two things have to happen.
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Corresponding angles have to be congruent, and corresponding sides have to be proportional.
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When we looked at congruent polygons, corresponding angles had to be congruent, and corresponding sides had to be congruent.
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But not for similarity: for similar polygons, angles have to be congruent, and sides have to be proportional.
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And polygons are only made up of angles and sides.
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Angles are congruent, and sides are proportional--these two things are like a checklist.
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These two things are what make up similar polygons: all corresponding angles have to be congruent, and all sides have to be proportional.
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Again, angles are congruent, and sides are proportional; and then those polygons will be similar.
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Let's look at these quadrilaterals here: quadrilateral ABCD and quadrilateral EFGH.
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Now, for these to be similar, I have to say that angle A is congruent to angle E, B is congruent to angle F,
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C is congruent to G, and D is congruent to H; and then, the sides are being proportional...
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remember, we talked about proportions in the previous lesson.
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So, if this, let's say, is 3, and this is 2, and I say that this is 6, then what does this have to be?
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If this is 3, and this is 2, then proportionally, if this is 6, then this has to be 4,
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meaning that the sides have to have the same ratio between corresponding sides.
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So, if this side to this side is 3:2, then AB to EF has to be 3:2; it has to be an equivalent ratio.
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We are actually going to talk about that in the next slide; it is called scale factor.
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As long as all of the sides are proportional, and all of the angles are congruent, we can say that they are similar.
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Now, the symbol for similarity is like that; it is a little squiggly, and that is it.
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Remember, for "congruent," you have a squiggly with the equals sign; this is without the equals sign, because they are not equal.
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It is only this--this means similar.
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So again, back to scale factor: it is the ratio of the lengths of two corresponding sides of two similar polygons.
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Here we have two similar polygons; now, I didn't show you with the angles that the angles are congruent;
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but just having this symbol right here tells you that these two polygons are similar.
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And therefore, we can assume that all corresponding angles (meaning angle A to angle E, B to F, C to G, and D to H) are all congruent.
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And all of the corresponding sides, like AB to EF, are going to be proportional,
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meaning that each pair of corresponding sides is going to have the same ratio.
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That ratio, then (once we know that it is similar), which must be the same (because, remember:
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the sides have to be proportional for them to be similar), is called the **scale factor**.
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So, if I asked you for the scale factor between quadrilateral ABCD and quadrilateral EFGH,
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you can take any two sides that are corresponding (let's say AB and EF) and make a ratio out of them.
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It will be 6:9, but then I have to simplify, so it becomes 2:3.
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So then, the scale factor is 2:3; now, let's look at the other side.
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4--the corresponding side to this is this, 6; so then, what is the ratio there? 4 to 6, which is 2:3.
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So, if they are similar, the ratio between all of the corresponding sides is going to be the same.
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And that ratio is called the scale factor, usually written as a fraction, like that; but it is just a ratio, 2:3, or 2/3; that is called the scale factor.
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Oh, and then, if I ask you for the scale factor of the quadrilateral EFGH and the quadrilateral ABCD, then it is no longer 2:3.
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Because I asked you for the scale factor of this one to this one, I have to say it is 3:2,
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because now the ratio you are going to make is going to be 9/6, so it is going to be 3:2.
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Let's work on our examples: Determine if each pair of figures is similar.
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Now, the first pair: we have triangles; remember the two things.
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#1: Angles have to be congruent; and then, what was #2?--sides are congruent?--no, sides have to be proportional.
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So, here let's see angle A and angle D; they are corresponding, and they are congruent.
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The next angle: corresponding angles are congruent, and this angle to this angle is congruent, so the first one checks; that one works.
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The next one, the sides being proportional: let's see, that means, because we are trying
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to determine if they are similar, we can't assume that they are all going to have the same scale factor.
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That is what we are checking for; so we have to find the ratio of each pair of corresponding sides.
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Then, the first pair, AB to DE: we know that they are corresponding, and we only know
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that they are corresponding because that is between A with the D and the angle B with E.
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So, AB would be DE; they are corresponding to each other.
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So then, the ratio between them is going to be 10:8, which is 5:4.
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So then, so far, we have this ratio between the sides; that means that all of the sides,
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each of the pairs of corresponding sides, should have that same ratio.
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If they do, then they are all proportional; if they don't, even if one pair of sides is wrong, then it is not similar, because all of the sides have to be proportional.
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So then, AC to DF is 5:4; look, it is the same--so far, so good.
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And then, BC to EF: BC is 7; EF is 6; wait a minute--is there any way that this and this are the same?--
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no, so this one is not the same; therefore, this one is "no," and these two triangles are not similar.
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So then, this answer would be "no" or "not similar."
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Now, the next pair, these two: let's see, A is congruent and corresponding to E, B with F, C with G, and D with H.
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Each pair of corresponding sides checks; and then, look at the other sides.
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Well, we know that BC is congruent to EF; but what do we know about BC with its corresponding side?
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The corresponding side of BC is FG; we don't know what this is--all that we know is that BC is congruent to EF.
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Well, that doesn't really tell us much, because, remember: it has to be proportional to the corresponding sides.
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If this is congruent to this, it just means that they so happen to have the same length.
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That doesn't really say much; that means that if this is 10, then this is also 10.
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But it doesn't matter, because this side has to be proportional to this, or the scale factor between this and this has to be equal to this to this.
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So, since we don't know, we don't have enough information to determine that the sides are proportional;
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so then, this would be "no"; and therefore, this is not similar.
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Here, for each pair of similar polygons, find the values of x and y.
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Now, if we look at the first pair, they are not both facing upright, but it is OK.
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Remember: you have to look at the angles to determine which sides are corresponding to what and what angles are corresponding to what.
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Angle A is congruent to angle E; angle B is congruent to angle D; and angle C is congruent to angle F.
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Those are corresponding angles; so here, we know that the corresponding angles are congruent; this one is true.
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And then, we have to look at their sides.
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Oh, I'm sorry; we are just finding the values of x and y, so we know that they are similar.
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Then I can go ahead and write "similar" like that, the symbol for being similar, because they tell us that it is similar.
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All I have to do here: since they are similar, I can assume that all corresponding sides are congruent.
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Then, I just have to solve for x and y, using that information.
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Let's see, x is AC; what is corresponding to AC?
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Well, A is congruent to E, and C is congruent to F, so AC is corresponding with EF.
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That means, remember, that I am not making them congruent, because sides are not congruent; sides are proportional.
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So, I am going to have to make a proportion, x/8; that is the ratio--these are corresponding: x/8.
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And that is going to be...I am going to write this in a different color, because it is a little hard to see...x/8; that is the ratio.
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And then, remember, since each pair of corresponding sides is going to have the same scale factor, I can use any other pair of sides that are given.
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So, x/8 is going to be an equivalent ratio to BC, which is 16, to DF, so 16/12.
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Now, I can simplify this, if I want; or I can just go ahead and solve it out.
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If I simplify, this is going to become 4/3; so if I just pretend that that is not there
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(I am just going to use these numbers, because they are smaller) then I can cross-multiply and use my cross-products.
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That becomes 3x = 32; well, then, x = 32/3, and that is the value of x.
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And you can leave it as a fraction.
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So again, let me explain that again: x:8 equals 0972; and then, the way I got 4/3 is just by simplifying this:
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16/12, if you just simplify that, becomes 4/3; and then, I use my cross-product: 3x =...4 times 8 is 32; divide the 3; x is 32/3.
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Now, to find y, I am going to do the same thing.
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Now, I know that x is 32/3, but that is a fraction, so I am going to use the same ratio here.
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So, y + 2, over...what is the corresponding side here?...it would be y + 2 to 15; that would be the scale factor.
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That is equal to, again, 4:3; then I use my cross-product, so that becomes 3y + 6 = 15(4); that is 60.
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I am going to subtract the 6, so 3y = 54; divide the 3; y =...let's see, that is 18.
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x is here, and y is there, for this first one.
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And again, you are just using proportions; you are just finding the ratio between a side and a corresponding side,
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and then a side and a corresponding side, another pair of corresponding sides.
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Here is the next one: now, I know this looks kind of confusing and complicated, but it is really not.
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It is just saying that this is a regular polygon; remember: a regular polygon is a polygon with all sides congruent and all angles congruent.
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So, it is equilateral, and it is equiangular; the same thing here--this is equilateral, and it is equiangular.
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Now, all of these angles are equal to all of these angles.
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Now, the sides: we know that these are proportional, so if this is 2, what is the scale factor between this one and this one?
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Well, this one is 2; what would be the scale factor to this?
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It would be 2 to 3; that is the scale factor, 2:3.
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That means that I can use this when I am solving for x and y.
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Here is x, and here is y; now, with this problem, because it is equilateral, you can actually take a shortcut.
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You can just make y - 1 equal to 2, because these sides are congruent; that would be the easier way to do it.
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Just do y - 1 = 2, and solve for y, and the same thing here: 9 - x = 3.
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Now, that is the shortcut way; but for the sake of doing proportions and solving it in this way, let's use a scale factor and make a proportion.
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y - 1, over (now, all of these are 3...this is 3; this is 3; they are all congruent) 3, equals...what is the scale factor? 2 to 3.
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So, this becomes 3y - 1 = 6; 3y = 7; did I do that right?
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Oh, I'm sorry; I didn't distribute it: 3y - 3 = 6, and then 3y = 9, so y = 3.
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And then, to find the x one: now, remember: if I am going to use 2/3, that means I have to state this side first, and then this side.
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So then, 2 over (if this is corresponding to this one, this is 2) 9 - x equals 2 over 3.
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So then, here I get...let's do this one first...2 times 9 is 18, minus 2x equals 6.
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How did you get that? You just cross-multiply and distribute; and then, -2x equals...
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If you subtract 18, that becomes -12; x = 6.
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So then, you have y equal to 3, and x equal to 6.
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The next one: Triangle ABC is similar to triangle EDC--that means A is congruent (remember, because angles are congruent--
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let me just write it out--"angles are congruent; sides are proportional)...
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I am just writing this; I know that I am writing this for every slide, but it is just so that you get used to seeing that angles are congruent; sides are proportional.
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Those are the two conditions which make polygons similar.
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Angle A is congruent to E; how do I know?--because A is written first, and E is written first.
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B is with D, and then C...we know that C is congruent to C; these are congruent to each other, because they are vertical angles.
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Find the measure of angle D; how are you going to find the measure of angle D?
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Well, I know that this triangle is going to add up to 180.
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So, if I find this angle, then I can find the measure of angle D, because they are the same.
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Here, 55 + 20 + x = 180; this becomes 75 + x = 180.
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Then, if I subtract 75, x = 105; that means that the measure of angle D is...this is 105; then this has to be 105, so it is equal to 105 degrees.
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Now, you find the scale factor; so then, because we know that they are similar, we can use any pair of corresponding sides to find the scale factor.
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Now, I can't use this pair, because it is not given; and I can't use CE with CA, because this one is not given; so I have to use BC to DC.
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They are asking for the scale factor between this triangle and this triangle, because that is the triangle that is listed first.
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The scale factor of triangle ABC to triangle EDC would be 9 to 10.
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And you can also write that as 9/10; that is the scale factor.
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Now, we have to find ED: this is the question mark.
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Since we know the scale factor, we can just use that, because remember: all of the sides are proportional.
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So, we can use this to find ED; that means 9/10, this to this, is going to equal...this is the corresponding side, so this is 4/x.
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To solve that out, cross-multiply: 9x = 40; x = 40/9, and that is ED.
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Example 4: Draw two similar figures for each--they are giving us the scale factor, and then we have to just draw whatever you want.
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So, your diagram is probably not going to look like mine, but it doesn't matter, as long as you draw two triangles with a scale factor of 2.
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Now, if the scale factor is 2, that is a whole number; remember: scale factors look like fractions.
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So, if a scale factor is a whole number of 2, I could turn that into a fraction, just by putting it over 1, so it would just be 2:1.
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Now, what does that mean? If the scale factor is 2:1, that means that, if one side of my first triangle,
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let's say, is 10, what would be the corresponding side of the other triangle?--it would be 5.
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So, this just means that my first triangle is going to be twice as big as my second triangle.
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Again, for similar polygons, they have the same shape, but a different size.
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So then, I need to have two triangles that have the same shape.
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You can't draw one triangle like this, and then another triangle, like a right triangle, if this is not a right triangle.
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So, make sure that they have the same shape.
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It is OK if you rotate it a little bit, and it is not positioned exactly like that.
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It is OK, but it just has to have the same shape.
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Let's say I have a triangle like that, and I have a triangle...
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Now, you know that you have to draw the first one bigger than the second one, because it is a scale factor of 2 to 1.
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So, I can just represent it like that, and then I can say 4 to 5...half of that would be 2.5, and then 6 and 3.
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This would be one example of two triangles that are similar, with a scale factor of 2.
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Look at the ratio of each of their sides; they are all going to be 2/1.
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Now, the next one is two quadrilaterals with a scale factor of 1 to 3.
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Remember: quadrilaterals--that means that, if one side of my quadrilateral is 1, then the second one has to be 3.
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So, you see how my second quadrilateral has to be bigger than my first quadrilateral.
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Let's say I want to draw a quadrilateral like this: my second quadrilateral is going to be a lot bigger.
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And mine was rectangles, but you can draw them as any type of quadrilateral; it just says "quadrilateral."
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So, show your angles, like this; your angles are congruent: 1, 2, 3, 4...
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And then, if this is 1, then this would be 3; if this is 1, this would be 3; if this is 3, then this will be 9;
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and the same thing here: 3 and 9--however you want to do it.
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Try to draw something a little bit different than this; make it a little more challenging.
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Just make sure that the scale factor is these two right here.
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That is it for this lesson; thank you for watching Educator.com.