Basic Math Similar Polygons

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

Polygons we know is some kind of shape.0008

If we have a triangle, triangles are polygons; squares, rectangles; those are all considered polygons.0013

Similar polygons means you have two polygons with the same shape.0022

They have to look exactly the same; but they are just different sizes.0029

One is going to be smaller or bigger than the other one.0036

But then they have to have the same exact shape.0039

When they are similar, it is a little symbol like this.0046

This means that this triangle here is similar to this triangle here.0049

It means that they have the exact same shape.0054

It means that one is not going to be any fatter and less taller and all that.0057

It is going to have the exact same shape.0063

But it is just going to be different sizes.0066

An example of similarities, if you are baby.0072

You are a baby; you have small hands; you have small feet; you are small.0079

As you get older, you grow; but everything has to grow proportionally.0085

Your hands grow and your feet grow the same amount.0090

If you are a baby and everything is small, as you grow older,0094

it is not like only your feet are going to grow but your hands stay the same size.0100

Everything has to grow according to how big and small or different let's say size is.0105

But then you are still going to have the same shape.0113

That is kind of an example of what it means to be similar.0115

Everything is proportional when things are similar.0119

Again if this is going to grow, if it is going to grow taller, then it also has to grow wider.0125

It has to grow in all areas just like a baby grows in all areas.0130

Again same shape but different size; then the corresponding sides are proportional.0137

Corresponding just means that the side that is basically related to each other.0144

This side and this side are called corresponding sides; corresponding sides.0152

It means that this side and this side are like the same.0160

They are being compared to each other.0164

Same thing here; this side with this side and this long side with this long side.0166

They are all corresponding.0171

That means I can create a ratio for each of these corresponding sides.0175

That means I can compare this one with this one.0182

4 to 6, remember that is a ratio; then it is all proportional.0185

Proportional means that this ratio is going to equal...0192

if I make a ratio for this, that is going to be the same.0195

For the third side too, this ratio to this is also going to be the same.0200

Just saying that all the sides, if you compare this side to this side,0207

that ratio is going to be the same as this side to this side.0211

It is also going to be the same as this side to this side.0214

We have three ratios; we only need two to make a proportion.0218

If you have a triangle, you are going to have three different ratios.0227

But you only need two.0229

You are only going to use the sides that they give you measures for.0232

Then you can create a proportion to solve for the missing side.0238

See how this all equals each other?--4/6 is equal to 4/6.0244

It is also equal to 6/9 because they all equal the same ratio of 2/3.0251

All of these ratios equal 2/3; that means these are all the same.0260

The first example is these two similar triangles.0268

You can draw a little similar symbol like that.0276

That means this triangle and this triangle have the same shape but just different size.0278

That means I can write a proportion and then find the value of X.0287

Here this side is corresponding with this side.0292

I can create a ratio comparing this to this.0300

The ratio will be 5 to X.0305

Again I want to write my ratio as a fraction because that is how I am going to solve my proportion.0311

This side to this side is 5 to X.0316

That means I can also create a ratio from this side to this side.0318

That will be 2 to 4.0323

Be careful, if you are going to make a ratio this to this,0327

then for the next ratio, the top number has to be from the same triangle.0333

If it is going to be this to this, then you have to make the next ratio this to that.0338

If you switch it around, then it is not going to be the same.0343

It is like saying boys to girls equals girls to boys.0346

You are flipping them; you are changing them; you can't do that.0353

If it is this triangle to that triangle, then your next ratio has to also be from this triangle to that triangle.0356

To solve this, you can use cross products.0364

Remember cross products is when you multiply across.0369

Or you can just simplify it and then use just mental math.0372

Here 2/4, this is the same as 1/2; how do I know?0378

2 divided 2 is 1; 4 divided by 2 is 2.0385

I can just make this also equal to 1/2.0390

1/2, that means the bottom number has to be double the top number.0397

5 over what?--what is X going to be?0401

If you multiply this by 5, you are going to get 5.0405

You have to multiply this by 5; you are going to get 10.0407

X has to equal 10; that means this side has a measure of 10.0412

Same thing here, we are going to write a proportion to find the value of X.0425

Here I can say this to this equal to this side to this side.0432

Or if I want, I can start off with this rectangle first as long as I stick to it for my second ratio.0442

5, corresponding side is X; 5/X equals... stick with the same one first... 7/14.0451

You can write it like that; or you can start with this one first.0466

It doesn't matter as long as you stick to that order.0469

7/14 is 1/2 because 7 divided by 7 is 1.0475

14 divided by 7 is 2.0484

That means I need to turn this also into 1/2.0488

1 times 5 is 5; 2 times 5 is 10.0493

X is going to equal 10.0503

If you want to practice cross products, again you are going to just do0510

5 times 14 which is going to be equal to X times 7.0514

I can write 7 times X.0524

You are going to just solve that out and then divide the 7.0527

You are going to solve for X that way.0532

You are still going to get 10.0533

70, 7 times 10 is going to equal 70.0536

For the third example, this is called a parallelogram.0546

It is not a rectangle because it is not perfectly going straight up and straight across.0554

It is not perpendicular; it is kind of tilting off to the side.0559

This is a parallelogram; but these are similar polygons.0564

Here this is corresponding with this side; this is corresponding with this side.0572

But they give you the other sides.0581

For a parallelogram, this side and this side are the same.0584

I can just write this as 12.0589

This side and this side are the same; this is going to be X.0592

When I write my proportion, I am just going to do the same thing.0597

Ratio of this to this side is 6 to X which is equal to 9 to 12.0602

Again I can figure out an equivalent ratio.0613

9/12 is the same as... let's divide this by 3; divide this by 3.0619

9 divided by 3 is 3/4.0626

That means this also has to be the same as 3/4.0630

3 times 2 equals 6; that means I have to multiply the 4 times 2.0638

X is going to give you 8; that means this side right here is 8.0645

Again you can just do cross product; 6 times 12 equals 9 times X.0652

Solve it that way.0660

For the fourth example, they give us a word problem.0664

We have to draw our own similar polygons.0670

A tree casts a shadow that is 10 feet long.0676

Let's see, I want to draw a tree; there is a tree.0681

I know my drawing is kind of bad; there is the ground; tree.0688

The shadow... let's say this is a shadow... is 10 feet long; this is 10 feet.0696

A person 5 feet tall is standing next to the tree.0708

Let's say the person is right here; draw a stick man.0713

This is still the same ground.0720

Person 5 feet tall is standing next to the tree and is casting a shadow.0722

Or let's say this person is 5 feet tall.0727

From here down to the ground is 5 feet.0731

Where this person is standing, his shadow is 3 feet.0737

The triangle formed by the person's height in the shadow...0747

That means height and shadow; this is a triangle; you can see that.0751

This triangle is similar to the tree and its shadow.0761

Then the triangle formed by this tree, here all the way down to this shadow.0767

These two triangles, this triangle here and this triangle here, are similar.0778

They want us to find... what is it?... the height of the tree.0785

How tall is the tree?--I am going to make this X, from here to here.0796

Because they said it is similar, I can make a proportion now.0803

I can say the 10 feet, the shadow, over the 3 because this side is corresponding to this side.0808

It is going to be equal to the tree's height.0820

Remember if you started off with this tree triangle, then you have to start it with the next one.0823

The tree height X over the person's height, 5.0830

From here, now it is a proportion; now I can just solve it out.0838

In this case, I can't simplify this.0845

I can't do the equivalent fraction method because this is already simplified.0847

There is no number that goes into both 10 and 3.0851

In this case, I just have to use cross products.0855

Here I want to do 3 times X.0862

3 times X equals 10 times 5 which is 50.0865

Again if I am going to solve for X, I need to divide this 3 because 3 times X is 50.0874

It is 50 divided by 3 to find the X.0880

If I want to find this, I have to do that.0887

Make sure this top number goes inside.0892

3 goes into 5 one time; 3, if I subtract it, I get 2; 0.0895

3 goes into 20 six times which is 18; I get 2.0903

Now that I have a remainder, I have to put my decimal point.0912

Bring down another 0; 3 goes into 20 again eight times.0918

18 again; 2; another 0; 8.0926

It depends on how many numbers after the decimal point your teacher wants.0934

But otherwise you can just probably leave it as 16.89.0940

Or maybe 16.9 if we are going to round this; round this from that number.0947

16.9; that will be in feet.0954

The X or the tree is 16.9, almost 17 feet tall.0961

Again create your proportion.0969

Make sure when you do your ratio, you are going to stick with the same side first.0970

It is this side to this side is your ratio.0976

Equals this side to that side ratio.0979

Then you just solve your proportion using cross products.0983

That is it for this lesson; thank you for watching Educator.com.0988