[ Silence ]
>> All right.
This is Part Two of solving more problems using proportions.
So here's an example.
On a map, two cities are two and five-eighths inches apart.
If three-eighths inches on the map represents 25 miles,
how far apart are the cities in miles?
All right.
We've all done this.
Right. You have to read a map, and they have a little scale
of miles, and you kind of have
to estimate how far something is.
So what we've got is really the distance on the map
and the distance in miles, or you could think of it
as the inches and the miles,
however you want to think about this.
So how about I put like what it's like on the map,
which is really the inches, right,
and then the actual distance, which happens to be in miles.
So somebody might write map over distance or some other way
to remember what we're putting in the numerator
and what we're putting in the denominator.
We're going to set a proportion up.
[ Demonstration ]
>> Okay. On a map two cities are two
and five-eighths inches apart.
And we don't know how far those cities really are
from each other; that's the question.
Right. We do know that three-eighths inches
on the map represents 25 miles.
Okay. So that's really helpful, three-eighths inches, 25 miles.
Okay. Now on the map, the two cities are really two
and five-eighths inches apart,
so two and five-eighths inches on the map.
And what we don't know is how far it's really apart in miles.
I'm going to put M for miles.
So this is one way of writing this proportion.
All right.
Now we can use --
[ Demonstration ]
>> -- cross products.
So I could have three-eighths times M --
[ Demonstration ]
>> -- equals --
[ Demonstration ]
>> -- 25 times two and five-eighths.
Now two and five-eighths --
it's going to be easier since I'm going to multiply fractions
to write that as an improper fraction.
So I'm going to write 25 as twenty-five over one, okay,
and two and five-eighths, I'm going to write that as --
well, to get 8 times 2 is 16 plus 5, 21 eighths.
Okay. Now I could go ahead and multiply 25 times 21,
but I always like to make my arithmetic easier.
So instead of multiplying that, I'm going to say how am I going
to solve this equation?
How am I going to isolate M?
I've got three-eighths times M. So how about if I multiply
by the reciprocal, eight-thirds, to both sides of the equation.
That's going to be nice because some canceling gets
to go on here.
The eights cancel, right, and also I can cancel that three
into the 21, which is seven,
makes my arithmetic a lot easier.
Over on the left side, it completely cancels, right,
the threes cancel, the eights cancel; so I'll be able to solve
for M. So I have 25 times 7, that's not so bad.
Think of quarters, seven quarters, 175 miles.
So M is 175, but we do have to answer this question.
So we say the cities are 175 miles apart.
And of course, we want to check to make sure
that seems reasonable, 175 miles apart, looking back up here,
three-eighths inches was 25 miles, so let's see, 175 miles,
that's seven times as much;
and so is three-eighths times 7 going to be two
and five-eighths, and actually it is.
So it's 7 times three-eighths, is how you would get
that two and five-eighths.
So it makes sense that 25 times 7 is 175 miles.
So that's our answer.
The cities are 175 miles apart.
Singular triangles have the same shape
but not necessarily the same size.
You could think of one being a magnification of the other.
So here's an example of two similar triangles.
They look the same, it's just that one is actually larger.
And in this particular set of similar triangles,
the sides of one triangle are actually twice as long
as the sides of the other triangle.
And in fact, that's what always happens when you're working
with similar triangles.
They're in proportion to each other.
So if you take the small side of one and you're comparing it
to the small side of the other, and it's one and a half times
as large, then all the other sides will also be one
and a half times as large and the sides
of the larger triangle.
So for instance, if I were going to compare the ratios here
from the small side to the --
the small side of the small triangle to the small side
of the large triangle, that would be the green sides;
so 4 over 8, and then let's compare the ratios
of the least sides, the small over the large.
We're going to put the numerator to be the small triangle.
It would be 11 over 22.
And if we look at the red side, it would be 9 over 18,
and notice these are all equal.
So we say they're, you know, they're proportional.
So they all actually reduce to one-half.
So we could use this idea to solve similar triangles.
So here's an example.
What if I said these are two similar triangles --
I'm telling you they're really --
just one's a magnification of the other --
and ask you to find this length, X. All right.
So we see which sides correspond here, hopefully.
[ Demonstration ]
>> So I put some color on this side
so we could see the least sides correspond
and the red sides correspond.
So I'm going to write a fraction.
And you could -- in the last example, I put like the length
of the small over the length of the large.
I'm going to do it the opposite this time.
I'll just put large over small
to show you it doesn't make any difference.
And we're trying to find X. All right.
So first, let's compare the least sides,
those similar sides, and I want to put the length
of the large side over the length of the -- I'm sorry --
the length on the small triangle compared to the length
on the small triangle, so that will be 12 over 5, okay,
and we say that it's going to be the same thing
as if I take the length for the red side, right,
on the large triangle and the length of the red side
of the small triangle, so that will be X over 6.
You have to be careful.
In the numerators, we're putting the length
of the large triangle, and you have to be consistent.
And so now we can solve this by using proportions.
So we have 5 times X, gets multiplied, and 12 times 6;
so that give me 5X equals 12 times 6
and then you'll have to divide by 5.
I wait until the end to multiply the 12 times 6,
but you don't have to.
So that's going to give you 72 fifths, right, which is 14
and two-fifths, or you could write 14.4.
So X, this side, is 14 and two-fifths.
And then you should look at it and make sure
that seems reasonable, just sort of look
at the picture and eyeball it.
Does that seems about right to you?
Well, it looks reasonable to me.
So then our answer is that it's 14 and two-fifths,
that's how long it is.
So that's just another application
where you can use proportions using -- on similar triangles.
Here's one for you to try.
So if the triangles below are similar,
find the length of the missing side.
All right.
So you see this missing long side on the big triangle.
So see if you could set up a proportion.
There's more than one way to do this one,
and see if you can get that length.
I'm going to call this X. So how about I do from small to large,
and I need to get X. So how about -- I could do 5 over X,
those correspond, 5 over X, and then I have
to choose either this 4 over 6, or 6 over 9.
It doesn't matter.
How about if I choose 4 and 6, do that proportion; 4, 6.
And now we can use proportions.
So I can write 4 times X equals 5 times 6, which is 30.
[ Demonstration ]
>> -- And if I reduce that, I get fifteen halves.
So that's seven and a half.
So the missing side is seven and a half.
Now look at it and see if that makes sense to you.
Yep. It's actually one and a half times this big.
You might've figured it out that way as well.
[ Silence ]