00:01
Find the perimeter of the shaded region.
00:03
So we have the shaded region and we want to find the perimeter of it and we have the arc bc that surrounds it, the segment xc and the segment bx.
00:13
So our perimeter would be the measure of arc bc plus the length of segment xc plus the length of segment bx.
00:27
And we have the arc of this semi, part of the arc is from a to b is 15 centimeters.
00:34
So let's focus on finding the arc bc first.
00:37
So we can see that if we take the measure of arc ab and add to that the measure of arc bc, that whole entire arc from here to here, that would give us the semicircle which is going to be one half of the circumference.
00:58
So one half of the circumference and the circumference would be two pi times the radius.
01:06
So that'd be one half times two pi r.
01:11
So the measure of arc ab, that's given to be 15 centimeters plus the measure of arc bc plus the one half and two would cancel and we're just left with pi and the radius is six.
01:24
We can see the radius is six centimeters.
01:26
So this would just be equal to pi times r or six times pi.
01:36
So we can solve for arc bc, the measure of arc bc.
01:40
We subtract 15 from both sides, we would get six pi minus 15.
01:45
And for now, we're going to leave that as an exact answer.
01:47
So now let's focus on finding xc.
01:55
How long is xc? well, we can see that xc plus ox, we add xc plus ox, that would give us oc, which again, oc is the radius, which would be six centimeters since every radii in the circle would be congruent.
02:21
So oc is six.
02:23
So we have xc then plus ox equals six.
02:32
What about ox? well, ox is, if we look at this right triangle that's formed here, ox is adjacent to this angle theta.
02:42
So we're going to need to find some information about angle theta before we can go any further.
02:47
So we can use arc formula for this.
02:53
The arc, any arc length is equal to the radius times angle theta if theta is in radians.
03:01
So we can solve for theta.
03:02
Theta would then be equal to s divided by r or the arc length divided by the radius.
03:08
So we know the arc length that's formed by this angle theta.
03:13
We know that arc length was bc, which we just found to be, s would be the measure of arc bc, which we have just found to be six pi minus 15.
03:25
So this would be s is six pi minus 15 divided by the radius is six...