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# geometry

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In the figure, triangle ABC is inscribed in a semicircle with diameter AC of length 20 inches, and AB = 10 inches. When the area of the shaded region, in square inches, is expressed in the form a*pi - b what is the value of a + b?

Jul 12, 2022

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Suppose O is the center of the circle. I am guessing that the shaded region refers to the area inside the sector BOA but outside of triangle ABO. But I am not sure if I am correct with what the shdaed region is equal to. If you get this wrong, don't blame it on me.

We know that BO = AO = 10, as they are all radii of the circle. Because AB is 10, we know that tringle ABO is equilateral, which also means that angle BOA is 60 degrees. This means that the sector is 1/6 of the WHOLE circle. The area of the whole circle is 100pi so the area of the sector is 50/3pi. We have to then subtract by the area of triangle ABO which is 25sqrt(3). Therefore a + b = 50/3pi + 25sqrt(3).

Jul 12, 2022
edited by Voldemort  Jul 12, 2022