Geometry Congruent and Similar Solids

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For the next lesson, we are going to go over congruent and similar solids.0001

Whenever we have two solids that are either similar or congruent, there is a scale factor.0010

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).0018

If we are going to use this side for the scale factor, then we have to use the corresponding side of the other solid.0028

So then again, the scale factor is the ratio of the two similar solids.0039

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 1:2.0046

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.0060

For congruent solids, these have to be true: the corresponding angles are congruent; the corresponding edges are congruent0072

(we have to have congruency between the two solids); the areas of the faces are congruent; and the volumes have to be congruent.0082

And congruent solids have the same size and same shape.0094

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.0100

Remember: whenever we have something similar, it has to be the same exact shape, but then just a different size.0113

Congruent solids will have the same shape and the same size.0120

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.0124

So, it is going to be a ratio of 1:1.0132

Looking at similar solids, if the scale factor is a:b, then the ratio of the surface areas is going to be a²:b²,0139

and the ratio of the volumes is going to be a³:b³.0153

Let's say we have two solids, and the scale factor between the two is 2:3;0161

then the ratio of the surface is going to be 2²:3², so it is going to be 4:9.0168

If that is the scale factor (that is the ratio between the two solids), their surface areas are going to be 4:9.0184

And then, the ratio of the volumes is going to be 2³:3³; 2 cubed is 8; 3 cubed is 27.0194

That is going to be the ratio of their volumes.0209

The first example is to determine whether each pair is similar, congruent, or neither.0218

Looking at these two, this pair: here, this is a cube, because we know that all of the sides are going to be congruent.0225

So, this is a cube; this is also a cube with all of the edges measuring 5.0241

So, in this case, because they are the same shape, but just different sizes, this is similar.0248

And we always know that all cubes are going to be similar, because cubes have the same shape.0258

No matter how big or how small, all cubes are the same shape; they can just be different sizes.0267

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.0273

But since they just have the same shape and different sizes, they are just similar.0282

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.0289

Remember: for these two to be congruent, they have to have congruent corresponding parts.0303

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,0315

because it is just the cylinder that is turned sideways; so if we say that that is the height,0320

then I need to find the height of this, so that I can compare.0327

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.0331

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.0344

It is going to be...if I name that h...h² + 24² is going to equal the hypotenuse (26) squared.0352

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.0365

This is 10; this is also 10; so then, their heights are congruent; their radius is congruent.0393

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).0402

Here, it is also going to be π times 12 squared; so the area of the bases will be the same.0414

To find the volume, it is going to be the area of the base...that is π, r, squared, times the height.0422

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.0432

So then, their volumes are going to be exactly the same.0444

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.0447

And again, because they are going to have the same volume, they are congruent.0460

Determine if each statement is true or false: All spheres are similar.0469

Spheres always have the same shape; no matter what, all spheres are the same shape.0477

Now, sizes could vary; we could have a large sphere; we could have a small sphere.0483

But they are always going to be similar, because they always have the same shape.0488

So, any time two solids have the same shape, they are always going to be similar; so this is true.0492

The next one: If two pyramids have square bases, then they must be similar.0504

Well, even if they have a square base, yes, squares are always similar, because squares always have the same shape.0511

A square is a square, whether it is large or small; they are always going to be similar.0524

But for pyramids, we can have a tall pyramid, or we can have a short pyramid.0531

So, it doesn't always mean that they are going to be similar--these do not have the same shape.0550

So, even if their square bases are exactly the same (they are congruent), because we don't know the height, this is false.0557

If two solids are congruent, then their volumes are equal.0572

Well, let's say that we have exactly the same rectangular prism; it is congruent to this.0578

If they are exactly the same, then isn't the space inside also going to be the same?0592

So, all of this is going to be the same as all of this; so this is true.0601

Congruent solids have congruent volumes, the same volume.0608

For this example, we are going to find the scale factor and the ratio of the surface areas and the volume.0617

The scale factor between this and this prism that are similar is going to be 4:6.0624

We are going to use corresponding parts to determine the scale factor; it is going to be 4:6.0634

We need to simplify this, and it is going to be 2:3; that is the scale factor between these two prisms.0641

Then, to find the ratio of the surface area, for surface area, it is a²:b².0649

And then, for volume, it is going to be a³:b³.0660

For surface area, the ratio is going to be 2 squared to 3 squared, which is 4 to 9.0668

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.0684

So, when we find the surface area, if we were to find the surface area of both this prism and this prism,0695

and then we simplify it, it is going to become 4:9.0701

And then, for volume, it will be 2³ to...what is that one, 3?...3³, so it is going to be 8:27.0710

And again, that does not mean that the volume of this is 8, and that the volume of this is 27.0727

Let's actually find the volume, given two corresponding sides right here.0736

This is similar, so the ratio between these two prisms is 3:2.0745

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.0755

You are naming this first; so it is this one to that one.0761

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 3:2.0765

Always keep in mind which one this is: this number refers to the larger prism.0771

Now, the volume of the smaller one, the second one, is given; it is 50 inches cubed.0779

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.0785

This is the larger one, over the smaller one; that means that, if I want to find the actual volume,0802

the volume of this one to the volume of this is going to become 27/8, simplified.0810

Then, I just know that I can make a proportion: this ratio is going to equal the volume of that0820

(because that one applies to the larger one), so let's say V for volume, over...what is the volume of this smaller one? 50.0827

That is how I make my proportion, because the volume of the larger to the volume of the smaller, simplified, is going to become 27/8.0838

Use the volume of the larger over the actual volume of the smaller prism.0848

So then, here I am going to solve out this proportion; this becomes 8V (cross-products: 8 times V) equals 27 times 50.0853

Using your calculator, 27 times 50 equals 1350; divide the 8; your volume is 168.75.0866

And then here, our units are inches (and for volume, it has to be) cubed--units cubed.0887

And that would be the volume of this larger prism.0897

OK, so again, to make your proportion, we know that this ratio has to equal this ratio.0901

They both are the ratios of their volumes; you have the volume of the larger prism to the volume of the smaller prism;0909

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.0923

You just make the two ratios equal to each other; set it equal.0934

Make sure that you keep the larger prism as your numerator; so it has to be 27 over the smaller prism.0939

And this is going to be V/50, the larger over the smaller.0949

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.0953

Find cross-products, and then just solve it out.0960

That is it for this lesson; thank you for watching Educator.com.0962