Hey ^-^ can someone please help me with this problem:
Find the area of the shaded regions below. Give your answer as a completely simplified exact value in terms of π (no approximations).

  8 (Pi - sqrt(3))

Discussion:

The area of the shaded region is that of the semicircle minus the area of the triangle..

Area of semicircle = 1/2 * Pi * R^2        

   Where R^2 is the square of the radius of the circle. In our case, R ( = OC)

    = 4 so the semicircle area is

   (1/2) * Pi * (4^2) = (1/2) * Pi * 16 = 8 Pi

Area of triangle.

  First of all, angle ACB is a right angle ( i.e. 90 degrees).

    * This is the Theorem of Thales from elementary Plane Geometry. *

 so by Pythagoras

   AC^2 + BC^2 = AB^2

But CB = 4 (given) and AB = 4*2 = 8 ( the diameter is twice the radius).

Substituting these in Pythagoras gives

   AC^2 + 4^2 = 8^2 or

   AC^2 = 8^2 - 4^2- = 64 - 16 = 48

   Hence AC = sqrt(48) = sqrt (16*3) = 4 * sqrt(3)

We are almost done! The area of the triangle is given by

  (1/2) b * h = (1/2)  BC * AC = (1/2) 4 * (4 * sqrt(3)) =  8 sqrt(3)

We conclude the area area of the shaded part is

 8 PI - 8 sqrt(3)   = 8 (Pi - sqrt(3))

Note that sqrt(3) is approx  1.7 so (PI - sqrt(3)) is a positive number, as it better well be!

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step-by-step explanation:

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