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We'll have to determine the length of the side of equilateral triangle to determine then it's area.
We know that the radius of the circle is of 10 cm.
If we'll join the center of the circle with 2 vertices of triangle, we'll get an isosceles triangle. Since the inscribed angle of equilateral triangle measures 60 degrees, then the corresponding central angle (which is the included angle between the two radii that are the legs of the isosceles triangle) measures 2*60 = 120 degrees.
We'll draw the height of the isosceles triangle that falls in the midpoint of the base of this triangle, base that is one of the legs of the equilateral triangle.
This height bisects the angle of 120. We'll get the right angle triangle, whose hypotenuse is the radius of circle.
We'll use the sine function to get the length of half of the leg of equilateral triangle:
sin 60 = x/10
`sqrt(3)` /2 = x/10
`sqrt(3)` = x/5
x = 5`sqrt(3)`
The length of each side of equilateral triangle measures 10`sqrt(3)` cm.
Now, we'll get the area of equilateral triangle;
A = 10`sqrt(3)` *10`sqrt(3)` *sin 60/2
A = 150`sqrt(3)` /2
A = 75`sqrt(3)` `cm^(2)`
Now, to get the area of the shaded region, we'll subtract the area of equilateral triangle from the area of the circumscribed circle.
The area of the circle is:
S = `pi` *`10^(2)`
S = 100 `pi` `cm^(2)`
Therefore, the area of the equilateral triangle is A = 75`sqrt(3)` `cm^(2)` and the area of the shaded region is S - A = (100`pi` - 75`sqrt(3)` ) `cm^(2)` .
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