Question 401186: An equilateral is inscribed in a circle of radius 10cm. Find the area of the equilateral.
Answer by edjones(8007)   (Show Source):
You can put this solution on YOUR website! Draw 3 lines from the center of the circle to bisect the 3 angles of the equilateral giving 3 congruent isosceles 30-30-120 triangles.
Find the base of one of the triangles
a/sin A = b/sin B
a/sin 120 = 10/sin 30
a/(sqrt(3)/2) = 10/(1/2)
2x/sqrt(3) = 20
2x=20sqrt(3)
x=10sqrt(3) a side of the equilateral (and a base of one of the isosceles).
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Bisect the apex angle of the quadrilateral dividing it into two 30-60-90 triangles, each with a base of 5sqrt(3).
30-60-90 triangles have a ratio of 1, sqrt(3), 2.
5sqrt(3) is the base so the altitude is 5sqrt(3)sqrt(3)=15
The area of of the equilateral is 1/2 * 15 * 10sqrt(3) = 75sqrt(3)
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Ed
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