WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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Now that we have gone over similar triangles, we are going to go over parts of those similar triangles for this lesson.
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Let's talk about their perimeters: now, we know that the perimeter is the sum of all of the sides.
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When it comes to these triangles, if they are similar, then their perimeters are going to have the same scale factor, the same ratio, as these corresponding sides.
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So, if, let's say, AB has a side of 8, and DE, the corresponding side, has a measure of 10;
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then the scale factor between these two triangles would be 8/10; or to simplify, it would be 4/5.
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So, this is the scale factor; that is the ratio.
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And then, since they are similar, we now know that their perimeters are going to have the same scale factor.
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The perimeters are going to be proportional to each other.
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The perimeter for this triangle--let's say it is 20: if the perimeter here is 20, then the perimeter here we can solve for, using the same ratio.
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So, if this is 4:5, then remember: I can make a proportion so that it will have the same ratio: 20/x.
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We do cross-products: 4x = 100; divide the 4; x is 25, so the perimeter of this triangle is 20, and the perimeter of this triangle is going to be 25.
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So again, as long as the triangles are similar, then the perimeters are going to have the same scale factor,
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or they are going to be proportionally the same as the corresponding sides.
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Now, in the same way, altitudes, remember, are segments that start from the vertex (or whose one endpoint is on the vertex),
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and they go down so that the other endpoint is going to be on one of the sides, so that the segment is perpendicular to that side.
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If you have a segment like that, and it is perpendicular, that is called an altitude.
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If you have an altitude drawn from a triangle, and an altitude from another triangle, and those two triangles are similar,
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then those altitudes are also going to have the same ratio, or they are also going to be equally proportional.
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They are going to be just the same; so here, if I give this point...say I label that P, and then this one I label N,
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then this altitude, BP, to the other altitude, EN, is going to have the same ratio, or scale factor, as any of the other sides.
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We already know that corresponding sides are going to be proportional, as long as the triangles are similar.
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So, we already know that AB to DE...that ratio is going to be the same as AC to DF and BC to EF.
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So then, they are all proportional; now we just say that those altitudes are going to be also proportional; they are going to have the same scale factor.
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So then, I can say BP/EN is going to be equal to...and then, any pair of corresponding sides, because it is going to be all the same.
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I can say that, since I named this altitude first, I have to name this triangle, so it is AB/DE.
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And again, for this one, I can name any corresponding sides; so BP/EN is equal to BC/EF, or AC/DF, and so on.
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That is altitude: a segment so that it is perpendicular to the side.
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And angle bisectors: an angle bisector is like an altitude, but then, it is not perpendicular.
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An angle bisector, remember, is something that cuts, in this case, an angle, in half; it bisects the angle.
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So again, it is starting from the vertex, going to the side opposite that vertex, so that the angle is bisected; that is the angle bisector.
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So again, if we have two triangles that are similar, then the angle bisectors are going to be proportional
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with the same scale factor as the corresponding sides of any one of those from the triangles.
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So, I can say that BN/EP is going to have the same scale factor as AB/DE.
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And the last one, the median--with the median also, the endpoints are on the vertex and the side opposite that vertex,
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so that it cuts that side into two equal parts, bisecting the side; that is the median.
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BN is a median of this triangle; EP is a median of this triangle; these two triangles are similar, so then,
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these medians are going to be proportional, with the same scale factor as any corresponding sides from those triangles.
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BN to EP, the other median, is going to be the same as AB to DE, or I can say equal to BC to EF, which is also equal to AC to DF.
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Now, make sure that, if you are going to name BN first, BN to EP, then for the second ratio, you have to name a part from this triangle first.
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So, you can't say BN to EP equals DE to AB; you can't say that--it has to be in the same order.
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Now, this one is a little bit different; it is not saying that the medians or the angle bisectors
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or the altitudes or the perimeters are all going to have the same proportional measure as the sides.
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This one is talking about the angle bisector; it is the angle bisector theorem where, if I have this angle bisector, CD,
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then this angle bisector is going to cut this triangle into proportional segments.
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It just means that, from here, I have AD and DB.
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So now, these parts are going to be proportional to the other two sides.
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To just read it, an angle bisector in a triangle separates the opposite side (which is this right here)
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into segments that have the same ratio as the other two sides.
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These two parts right here...AD over DB is going to be the same ratio as the other two sides, AC over BC.
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Again, AD, this part, to DB, equals AC to BC.
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This was a little bit different than what we just went over for angle bisectors,
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because that is saying, "Well, if I have two triangles and an angle bisector, then this angle bisector
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with this angle bisector is going to have the same ratio as the other sides,"
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because if you have two similar triangles, you have a scale factor between them,
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meaning the ratio between this triangle and this triangle.
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And it is just saying that those two different angle bisectors are going to have that same ratio between them.
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This one is different, because it is only giving you one angle bisector.
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It is not comparing an angle bisector with an angle bisector from another triangle.
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This one is one single angle bisector, and that one angle bisector is going to cut this one triangle up into different segments that are going to be proportional.
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So again, AD to DB: those segments are going to be proportional to AC to BC, the other two sides.
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That is the angle bisector theorem: there are two different theorems that we have gone over for the angle bisector.
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The first one: it is talking about two different angle bisectors; this one is one single bisector
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that is cutting this triangle up into different parts that are going to have the same ratio.
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Let's go over some examples: the first one...we are just going to use these two; this is actually #2 (let me write that in).
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The first one: there are two triangles, and they are similar.
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And this right here--those are medians, because they are bisecting the side right here.
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So then, there are two equal parts (segments); that is the median.
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And it doesn't matter, because they are all the same; it is just different theorems, but then, it is all the same concept--
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that they are all going to have the same ratio.
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I can say that 15, the first median, over...the second median is x...equals one side (that is 10), over another side (is 18).
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These are corresponding sides, so we can use that ratio.
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From here, now, I am going to do cross-products.
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But see here: 15 times 18--that is kind of a big number.
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I am going to simplify this; this is the same as 5/9.
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I can just use this; this is equal to this equivalent fraction; I am just going to use this.
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And it is just easier to just multiply 15 times 9.
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You can go ahead and use that; it is fine to multiply that; but I am just going to make it 5/9.
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Then, 15 times 9 is 135; 135 is equal to (that was this side right here) 5x.
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So, divide the 5, and you get 27; so that is equal to x.
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And then, #2: here, we have altitudes, this altitude and this altitude.
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And we know that it is an altitude because it is perpendicular to the third side.
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This one is not perpendicular; it is at the midpoint of the sides; so here, we have altitudes (let me just write that in).
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So then, this one here that we had...these are medians...and then, for this one, we have altitudes.
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Again, we are going to use the same strategy by doing altitude over altitude, equal to side over side.
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And then, here, I can also simplify this to make that 3/4; 3/4 = 12/x; 4 times 12 is 48; that is equal to 3x.
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Divide the 3; then I get 19 = x...no, that isn't 19; 16 = x.
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Here, we know that this median is going to be 27; and here, we know that this side is going to be 16.
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OK, the next one: Find the value of DC.
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There are a few things to look at here: We have an angle bisector, because of this; this shows me that it is bisecting this angle.
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And then, let's see, I don't have two similar triangles; I have that this side is 18; this little part right here is 4; BC is 14.
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I know that this is also 14, because these are congruent.
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I don't know what this side is, or this side; and this is the side I am looking for.
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So, let's see: what can we use here?
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We know that the angle bisector theorem says that, if we have an angle bisector...
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now, this angle bisector can be for this triangle, or it could be for the big triangle; so either way, here,
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it is going to be part to part equals part to part (the other two sides), or the angle bisector if I am looking at the big triangle--
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then it is going to be part to part equals side to side.
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But then, I don't really know what these parts are; I can't say that this is half exactly, because I don't know.
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This is not a median; if this was a median, then I know that this would be half of that measure, but it is not.
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Here, these two sides being congruent, and these two parts being congruent for this side--
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that gives me more information; that kind of helps me out.
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From the previous lesson, we know that, if I have a segment within a triangle,
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if I look at the big triangle, and then I think of this as a segment, and this endpoint is the midpoint of this side,
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and this endpoint is the midpoint of this side, then I know that this is the mid-segment.
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It is the middle segment so that this is parallel to EA, and it is half the measure of EA.
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Remember that? If DB is half the measure of EA, well, EA is 18; that means this whole thing right here is going to be 9.
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This DF is 4; what would that make FB? That would make that 5.
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Now, I am going to label that x; so, now I know that I can use my angle bisector theorem,
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because right here, if I look at the smaller triangle, I have this part; I know that part over part is going to equal side over side.
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And I have these parts, and I have this side, so only this side is missing.
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That means that I can go ahead and write my proportion: 4/5 (remember, part to part) is going to equal DC, or x, over BC, which is 14.
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So, from here, I am going to do cross-products: 14 times 4 is going to be 56; that equals 5 times DC.
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I am going to divide the 5, and then I know that DC is going to be 56/5.
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And you can just go ahead and leave it like that; so DC is 56/5.
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Or if you want to, you can just make it into a mixed number, which is 11 and 1/5, or 11.2.
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But then, that should be fine: DC is 56/5.
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The next one: BD and NP (here is BD, and here is NP) are medians.
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That means...well, they didn't show you that they are medians; they actually told you that they were medians.
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So then, you can just draw in these little slash marks, and then find the value of x.
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We know that medians are going to have the same ratio as the ratio between similar triangles.
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I can say that 11, the first median, over the second median, 10, is going to equal 2x - 1, over 3x - 4.
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Now, you are probably thinking, "Well, why is this ratio this part and this part--it is not the whole side?"
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The theorem before on the median said that median to median is equal to side to side.
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But this is not a whole side; it is just part of a side.
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But it would actually be the same thing, because they are medians; and let me just explain that.
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If this part is 2x - 1, this whole thing is going to be 2 times 2x - 1, because this is 2x - 1, and this is also 2x - 1.
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So, it is 2 times 2x - 1; well, this whole thing is going to be 2 times 3x - 4.
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So, if you want to use a theorem and say that 11, the median, over median, is equal to side,
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which is 2(2x - 1), over side, 2(3x - 4), what happens here?
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These 2's will cancel out, so it just becomes the same thing as that.
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And it is only like that, because they are medians; and then, those two sides are equal.
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From here, let's go ahead and solve it; cross-multiply; 11 times all that...let me just write it out.
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Then, I can distribute, so that becomes 33x - 44 = 20x - 10.
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If I subtract the 20x, that becomes 13x =...I am going to add the 44, so I get 34.
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And then, I divide the 13, so x is going to equal 34/13.
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I know that this is not going to be able to simplify; they don't have a common factor; so that is my answer, 34/13.
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Again, you can just leave it like that, or you can make it a decimal, or you can make it into a mixed number, if you would like.
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But that is the answer for x; so again, just make it into a proportion; just go ahead and cross-multiply and solve it out.
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The fourth example: Find the perimeter of triangle ABC.
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Here, all you are given is the length of AB and DE; and again, we are going to say that this is similar.
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The perimeter is triangle DEF is given; and then, we are going to look for the perimeter here.
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So then, the perimeter is what we are solving for.
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Remember the first theorem that we went over for this lesson; it said that the perimeters are going to have the same ratios as the triangles' corresponding sides.
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So, the ratio from AB to DE is going to be 10.2/12.5.
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It is going to equal...I am just going to write P for perimeter, and that is this triangle here...over 32, the perimeter of that one.
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OK, so then, you are going to multiply this.
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If you multiply...let's do that over here, and I will just multiply it out...this becomes 402; this becomes 603, 326.4...
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OK, let me just double-check: 402, 603, and then my decimal goes right there: 326...if you have a calculator,
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you can just use your calculator...that equals 12.5P.
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And then, you have to divide the 12.5, so it is all that divided by 12.5.
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And then, for this one, you wouldn't be able to leave it like that, because you have decimals within your fraction.
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When you have decimals within your fraction, you want to simplify.
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You want to either change your whole fraction to a decimal, or just simplify so that your decimals go away.
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326.4, divided by 12.5: this is going to be...and then, you can just go ahead and use your calculator.
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That is the easiest way to do this; for us here, we are going to say 2...this becomes 250;
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if I subtract that, it is going to become 76, with a 4; and then, let's see, 250, 500, 750...so then, let's say 6.
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We multiply this by 6, and I am going to get 30, 12, 15, 7; so then, 6...and then, here is the 0...and that is .1.
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So, I am just going to leave it like that...something around there.
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The perimeter is 26.1, and then you could just use your calculator for that; that is the perimeter.
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That is it for this lesson; thank you for watching Educator.com.