Hello and welcome to the section on similar figures. As you know, similar figures have the same shape, but may have different sizes.
And if two geometric figures are similar, then corresponding sides are in proportion. Let's go to the board now and work our first problem.
We have two similar triangles: triangle ABC is similar to triangle EFG. And in these two similar triangles, side AC corresponds to side EG,
side AB corresponds to side EF, and side CB corresponds to side GF. In other words,
the longest side in one corresponds to the longest side in the other, and the shortest side in one corresponds to the shortest side in the other.
Now, we want to find the length of side AB, so, I'm going to call that x. I'm going to call that length x, and I'm going to set up my proportion this way:
I'm going to say x is to 6 as 6 is to 4, so, x is to 6 as 6 is to 4.
So, I'm using that property I have for similar figures that says, if two figures are similar, then corresponding sides are in proportion,
which means that this side is to this side as this side is to this side. Okay, so now I have my proportion.
What I'm going to do is use my means-extremes property to solve this proportion. So, I'm going to say, x(4) = 4x; and then, 6(6) = 36.
(Okay, I'm going to move that camera down a little bit. There we go. Now, we have a little bit more room to work.)
And let's see, 4(x) = 36. If I divide both sides of this by 4, or multiply both sides by 1/4, I end up with x = 9.
So, in this problem right here, with these two similar triangles, if the two triangles are similar, then corresponding sides are in proportion,
that allows us to set up this proportion right here. We use the means-extremes property to multiply x(4); we get 4x; 6(6) = 36.
Then, we divide both sides by 4 to end up with x = 9. So, if these two triangles are similar, then it must be true that x = 9.
Let's take a look at our next problem. In this problem, I have two similar rectangles;
and I've drawn this width right here as 12; this length as 16; this width is 9, and this length is x.
And since these two rectangles are similar, the width in one is to the width in the other as the length in one is to the length in the other. So,
again, I'm going to just use the fact that corresponding sides of similar figures are proportional, and write my proportion this way:
x is to 16, so x/16 is equal to 9/12. So, what I'm saying is, this side is to this side, as this side is to this side;
or x is to 16 as 9 is to 12. So, that gives me a little proportion here to solve. And so, I'm gong use my means-extremes property:
x(12) = 12x; and then, 16(9) = 144. So, I have 12x = 144. I'll divide both sides by 12, and I'll end up with x = 12.
So, if these two rectangles right here are similar, then it must be true that x = 12 because this side is to this side as this side is to this side.
I set up my proportion. I used my means-extremes property to multiply x(12), and 169; that gives me 12x = 144.
And then, I solved that to end up with x = 12. So, again, if two figures are similar, then corresponding sides must be in proportion.
Let's look at one last problem. In this problem, we have a tree casts a shadow of 38 feet long when a 6 foot man casts a shadow 4 feet long. Now,
what you have to assume about a problem like this is that the rays from the sun are what caused the shadows to be whatever length they are;
and that the rays from the sun come down parallel to each other, so that we end up with two similar triangles, right here.
So, if that's the case, then we can find the height of this tree right here, using our property of similar figures
that says corresponding sides are in proportion. So, that means that the height of the tree is to the height of the man,
as the length of the shadow of the tree is to the length of the shadow of the man, or x is to 6 as 38 is to 4.
Now, I can use my means-extremes property to solve this by saying that x(4), which is 4x, must be equal to 6(38).
Now, I'll work that out on a calculator. That means that 4x must be equal to 228.
I divide both sides of this equation, then, by 4 to end up with, let's see, 228 ÷ 4 on the calculator is 57,
57; and I'm going to put my units in here, feet: x = 57 feet. So, if a tree casts a shadow 38 feet long
at the same time a 6 foot man casts a shadow of 4 feet long, then the height of the tree must be equal to 57 feet.
And we found this by using our property of similar figures that says corresponding sides are in proportion.