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In this diagram we have two arcs centered at point p.
Arc AB.
And you could imagine this arc could
be part of a circle like this that is centered at point AB.
And then you have this larger arc
cd that would be part of a larger circle, just like that,
centered at point P.
And we don't know the actual angle measure
of angle P. Right over here.
But without even knowing that, just
knowing the lengths of these segments here,
I want to think about whether we can figure out
the ratio of the length of arc AB, not angle AB,
to the length of arc-- I'll do that in green-- CD.
And I encourage you to pause this video now and think
about this on your own.
Well, they didn't give us the radian measure here,
or any type of angle measure here,
but let's just imagine that it is x radians.
We've already seen that one way to think about an angle
measure in terms of radians is that this
is a count of the number of the length of the arc
that it intercepts in terms of radii.
So if this has an angle measure of x radians
then that means that this arc length
is going to be x times the radius.
It's going to be x radii long.
The radii is 5 units long so this
is going to be 5-- I'll do that in that, this
is the right color-- this is going to be 5x units long.
Well, same argument for this arc length right over here.
It has a radius of 9.
5 plus 4.
And so this is going to be x radii long.
Each radius is 9 units.
So it's going to be 9x units long.
So what's the ratio of these two lengths?
Well, the smaller arc AB would have a length of 5x.
Arc CD would have a length of 9x.
So the ratio is 5 to 9.
Now let's say someone says, oh, that was pretty good,
you were able to figure out this ratio.
Let's say we were to give you one of these lengths.
Let's say that they were to tell you that the length of arc AB
is 25/8.
Given that, what else could you figure out about it?
Well, you could actually figure out
everything else you would want to know.
You could figure out the angle measure in terms of radians
and you could figure out the arc length of CD.
And I encourage you to pause the video now and figure them out
on your own.
Well, let's work through each of them.
We already figured out that ab's length is 5x.
So we could just say that 5x is going to be equal to 25/8.
Divide both sides by 5. x is equal to 5/8.
And so the measure of this angle right here is 5/8 radians.
So what's going to be the measure of this arc length?
So this is going to be 9 times x.
That's just 45/8.