Geometry Similar Polygons

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For this next lesson, we are going to go over similar polygons.0002

Now, polygons, we know, are shapes that have three or more sides.0006

They have to be enclosed, meaning that there can't be any open gaps; and remember that the sides can't overlap.0012

And the sides do have to be straight.0018

So then, we know that quadrilaterals are polygons; triangles, pentagons...all of those are polygons.0020

Now, we are going to talk about similarity; we know that congruent polygons are polygons that are exactly the same--0030

the same size and same shape; now, similar polygons are a little bit different.0045

They have the same shape, but they are not the same size.0051

Two polygons are similar if and only if their corresponding angles are congruent, and the measures of their corresponding sides are proportional.0057

In order for two polygons to be similar, two things have to happen.0069

Corresponding angles have to be congruent, and corresponding sides have to be proportional.0078

When we looked at congruent polygons, corresponding angles had to be congruent, and corresponding sides had to be congruent.0092

But not for similarity: for similar polygons, angles have to be congruent, and sides have to be proportional.0101

And polygons are only made up of angles and sides.0110

Angles are congruent, and sides are proportional--these two things are like a checklist.0118

These two things are what make up similar polygons: all corresponding angles have to be congruent, and all sides have to be proportional.0136

Again, angles are congruent, and sides are proportional; and then those polygons will be similar.0146

Let's look at these quadrilaterals here: quadrilateral ABCD and quadrilateral EFGH.0153

Now, for these to be similar, I have to say that angle A is congruent to angle E, B is congruent to angle F,0161

C is congruent to G, and D is congruent to H; and then, the sides are being proportional...0172

remember, we talked about proportions in the previous lesson.0183

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?0186

If this is 3, and this is 2, then proportionally, if this is 6, then this has to be 4,0206

meaning that the sides have to have the same ratio between corresponding sides.0215

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.0220

We are actually going to talk about that in the next slide; it is called scale factor.0231

As long as all of the sides are proportional, and all of the angles are congruent, we can say that they are similar.0237

Now, the symbol for similarity is like that; it is a little squiggly, and that is it.0243

Remember, for "congruent," you have a squiggly with the equals sign; this is without the equals sign, because they are not equal.0251

It is only this--this means similar.0260

So again, back to scale factor: it is the ratio of the lengths of two corresponding sides of two similar polygons.0268

Here we have two similar polygons; now, I didn't show you with the angles that the angles are congruent;0275

but just having this symbol right here tells you that these two polygons are similar.0281

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.0287

And all of the corresponding sides, like AB to EF, are going to be proportional,0300

meaning that each pair of corresponding sides is going to have the same ratio.0308

That ratio, then (once we know that it is similar), which must be the same (because, remember:0313

the sides have to be proportional for them to be similar), is called the scale factor.0322

So, if I asked you for the scale factor between quadrilateral ABCD and quadrilateral EFGH,0329

you can take any two sides that are corresponding (let's say AB and EF) and make a ratio out of them.0337

It will be 6:9, but then I have to simplify, so it becomes 2:3.0345

So then, the scale factor is 2:3; now, let's look at the other side.0355

4--the corresponding side to this is this, 6; so then, what is the ratio there? 4 to 6, which is 2:3.0362

So, if they are similar, the ratio between all of the corresponding sides is going to be the same.0373

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.0381

Oh, and then, if I ask you for the scale factor of the quadrilateral EFGH and the quadrilateral ABCD, then it is no longer 2:3.0397

Because I asked you for the scale factor of this one to this one, I have to say it is 3:2,0405

because now the ratio you are going to make is going to be 9/6, so it is going to be 3:2.0414

Let's work on our examples: Determine if each pair of figures is similar.0425

Now, the first pair: we have triangles; remember the two things.0431

#1: Angles have to be congruent; and then, what was #2?--sides are congruent?--no, sides have to be proportional.0439

So, here let's see angle A and angle D; they are corresponding, and they are congruent.0462

The next angle: corresponding angles are congruent, and this angle to this angle is congruent, so the first one checks; that one works.0471

The next one, the sides being proportional: let's see, that means, because we are trying0483

to determine if they are similar, we can't assume that they are all going to have the same scale factor.0489

That is what we are checking for; so we have to find the ratio of each pair of corresponding sides.0494

Then, the first pair, AB to DE: we know that they are corresponding, and we only know0499

that they are corresponding because that is between A with the D and the angle B with E.0507

So, AB would be DE; they are corresponding to each other.0513

So then, the ratio between them is going to be 10:8, which is 5:4.0517

So then, so far, we have this ratio between the sides; that means that all of the sides,0525

each of the pairs of corresponding sides, should have that same ratio.0531

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.0536

So then, AC to DF is 5:4; look, it is the same--so far, so good.0548

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?--0560

no, so this one is not the same; therefore, this one is "no," and these two triangles are not similar.0572

So then, this answer would be "no" or "not similar."0582

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.0593

Each pair of corresponding sides checks; and then, look at the other sides.0610

Well, we know that BC is congruent to EF; but what do we know about BC with its corresponding side?0619

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.0628

Well, that doesn't really tell us much, because, remember: it has to be proportional to the corresponding sides.0640

If this is congruent to this, it just means that they so happen to have the same length.0648

That doesn't really say much; that means that if this is 10, then this is also 10.0655

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.0661

So, since we don't know, we don't have enough information to determine that the sides are proportional;0674

so then, this would be "no"; and therefore, this is not similar.0681

Here, for each pair of similar polygons, find the values of x and y.0695

Now, if we look at the first pair, they are not both facing upright, but it is OK.0703

Remember: you have to look at the angles to determine which sides are corresponding to what and what angles are corresponding to what.0714

Angle A is congruent to angle E; angle B is congruent to angle D; and angle C is congruent to angle F.0722

Those are corresponding angles; so here, we know that the corresponding angles are congruent; this one is true.0734

And then, we have to look at their sides.0745

Oh, I'm sorry; we are just finding the values of x and y, so we know that they are similar.0749

Then I can go ahead and write "similar" like that, the symbol for being similar, because they tell us that it is similar.0753

All I have to do here: since they are similar, I can assume that all corresponding sides are congruent.0763

Then, I just have to solve for x and y, using that information.0771

Let's see, x is AC; what is corresponding to AC?0775

Well, A is congruent to E, and C is congruent to F, so AC is corresponding with EF.0782

That means, remember, that I am not making them congruent, because sides are not congruent; sides are proportional.0792

So, I am going to have to make a proportion, x/8; that is the ratio--these are corresponding: x/8.0797

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.0808

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.0819

So, x/8 is going to be an equivalent ratio to BC, which is 16, to DF, so 16/12.0830

Now, I can simplify this, if I want; or I can just go ahead and solve it out.0845

If I simplify, this is going to become 4/3; so if I just pretend that that is not there0851

(I am just going to use these numbers, because they are smaller) then I can cross-multiply and use my cross-products.0860

That becomes 3x = 32; well, then, x = 32/3, and that is the value of x.0866

And you can leave it as a fraction.0883

So again, let me explain that again: x:8 equals 0972; and then, the way I got 4/3 is just by simplifying this:0885

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 32/3.0899

Now, to find y, I am going to do the same thing.0914

Now, I know that x is 32/3, but that is a fraction, so I am going to use the same ratio here.0919

So, y + 2, over...what is the corresponding side here?...it would be y + 2 to 15; that would be the scale factor.0926

That is equal to, again, 4:3; then I use my cross-product, so that becomes 3y + 6 = 15(4); that is 60.0943

I am going to subtract the 6, so 3y = 54; divide the 3; y =...let's see, that is 18.0960

x is here, and y is there, for this first one.0981

And again, you are just using proportions; you are just finding the ratio between a side and a corresponding side,0985

and then a side and a corresponding side, another pair of corresponding sides.0991

Here is the next one: now, I know this looks kind of confusing and complicated, but it is really not.0996

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.1005

So, it is equilateral, and it is equiangular; the same thing here--this is equilateral, and it is equiangular.1018

Now, all of these angles are equal to all of these angles.1025

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?1029

Well, this one is 2; what would be the scale factor to this?1038

It would be 2 to 3; that is the scale factor, 2:3.1042

That means that I can use this when I am solving for x and y.1050

Here is x, and here is y; now, with this problem, because it is equilateral, you can actually take a shortcut.1054

You can just make y - 1 equal to 2, because these sides are congruent; that would be the easier way to do it.1064

Just do y - 1 = 2, and solve for y, and the same thing here: 9 - x = 3.1073

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.1080

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 3.1093

So, this becomes 3y - 1 = 6; 3y = 7; did I do that right?1112

Oh, I'm sorry; I didn't distribute it: 3y - 3 = 6, and then 3y = 9, so y = 3.1126

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.1143

So then, 2 over (if this is corresponding to this one, this is 2) 9 - x equals 2 over 3.1152

So then, here I get...let's do this one first...2 times 9 is 18, minus 2x equals 6.1168

How did you get that? You just cross-multiply and distribute; and then, -2x equals...1177

If you subtract 18, that becomes -12; x = 6.1184

So then, you have y equal to 3, and x equal to 6.1192

The next one: Triangle ABC is similar to triangle EDC--that means A is congruent (remember, because angles are congruent--1198

let me just write it out--"angles are congruent; sides are proportional)...1215

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.1230

Those are the two conditions which make polygons similar.1239

Angle A is congruent to E; how do I know?--because A is written first, and E is written first.1245

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.1254

Find the measure of angle D; how are you going to find the measure of angle D?1268

Well, I know that this triangle is going to add up to 180.1273

So, if I find this angle, then I can find the measure of angle D, because they are the same.1280

Here, 55 + 20 + x = 180; this becomes 75 + x = 180.1285

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.1307

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.1322

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.1332

They are asking for the scale factor between this triangle and this triangle, because that is the triangle that is listed first.1345

The scale factor of triangle ABC to triangle EDC would be 9 to 10.1356

And you can also write that as 9/10; that is the scale factor.1365

Now, we have to find ED: this is the question mark.1371

Since we know the scale factor, we can just use that, because remember: all of the sides are proportional.1377

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.1383

To solve that out, cross-multiply: 9x = 40; x = 40/9, and that is ED.1402

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.1418

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 2.1430

Now, if the scale factor is 2, that is a whole number; remember: scale factors look like fractions.1439

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 2:1.1447

Now, what does that mean? If the scale factor is 2:1, that means that, if one side of my first triangle,1459

let's say, is 10, what would be the corresponding side of the other triangle?--it would be 5.1470

So, this just means that my first triangle is going to be twice as big as my second triangle.1477

Again, for similar polygons, they have the same shape, but a different size.1489

So then, I need to have two triangles that have the same shape.1500

You can't draw one triangle like this, and then another triangle, like a right triangle, if this is not a right triangle.1504

So, make sure that they have the same shape.1511

It is OK if you rotate it a little bit, and it is not positioned exactly like that.1513

It is OK, but it just has to have the same shape.1519

Let's say I have a triangle like that, and I have a triangle...1525

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 1.1527

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 3.1535

This would be one example of two triangles that are similar, with a scale factor of 2.1562

Look at the ratio of each of their sides; they are all going to be 2/1.1569

Now, the next one is two quadrilaterals with a scale factor of 1 to 3.1575

Remember: quadrilaterals--that means that, if one side of my quadrilateral is 1, then the second one has to be 3.1583

So, you see how my second quadrilateral has to be bigger than my first quadrilateral.1593

Let's say I want to draw a quadrilateral like this: my second quadrilateral is going to be a lot bigger.1599

And mine was rectangles, but you can draw them as any type of quadrilateral; it just says "quadrilateral."1611

So, show your angles, like this; your angles are congruent: 1, 2, 3, 4...1619

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;1633

and the same thing here: 3 and 9--however you want to do it.1644

Try to draw something a little bit different than this; make it a little more challenging.1656

Just make sure that the scale factor is these two right here.1662

That is it for this lesson; thank you for watching Educator.com.1670