SOLUTION: In the diagram, O is the centre of the circle and sector angle COB is 120 degrees. CB has arc length 4 π cm. Also AC=AB. Find the area, in cm*2, of the shaded region. A)30 B)18

Question 1166971: In the diagram, O is the centre of the circle and sector angle COB is 120 degrees. CB has arc length 4 π cm. Also AC=AB. Find the area, in cm*2, of the shaded region.
A)30
B)18 √ 3
C) 27 √ 3
D) 27
E) 24 √ 3
https://imageshack.com/i/pn2akcrPj

Found 2 solutions by Boreal, ankor@dixie-net.com:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
It is B. We know the radius is 6 cm because the 120 degree arc is 4π so the perimeter is 12π. C=2πr
the shaded area may be divided in half by drawing the radius through the center. This gives rise to two larger triangles. The angle at the vertex was originally 60 degrees, because the arc is 120 degrees. That makes the new triangles 30-30-150 and the two right triangles made from it are 30-60-90 with the altitude being 3, because the hypotenuse is the radius.
If the altitude is 3, the adjacent side is 3 sqrt (3)
The area of one of those triangles is (1/2)bh, so the area of all 4 of them, 2 on bottom and two on top, is 2(3*3sqrt(3), and that is 18 sqrt (3)
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
In the diagram, O is the centre of the circle and sector angle COB is 120 degrees.
CB has arc length 4 π cm.
Also AC=AB. Find the area, in cm*2, of the shaded region.
:
Find the radius, the given arc = 4 and is 1/3 the circumference therefore

divide both sides by pi
2r = 12
r = 6 cm is the radius
:
draw the radius from A to 0 which gives us two equal isosceles triangles
Each of these triangle can be divided into two right triangles with angles of 60 degrees at the center.
The hypotenuse of these triangles is the radius, 6 cm
Find the other two side of the right triangles using sine and cosine of 60 degrees.
sin(60) = s1/6
s1 = 5.196 cm
cos(60) = s2/6
s2 = 3 cm
:
Find the area of one of the right triangles
A = 5.196 * 3
A = 7.794 sq/cm
The shaded are consist of 4 right triangle, therefore
4 * 7.794 = 31.176 sq/cm is the shaded area which is B
:

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