The diagram above shows a circle inscribed inside a square of side length
, which has a right triangle of legs length
and
, and hypotenuse length
inscribed in it.
The difference between the area of the shaded region and the shaded region can be written as for integers and . Let the digit sum of be . Then find the remainder of
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Since the triangle is inscribed in the circle then it's hypotenuse is also the diameter of the cricle so y = 1 0 .
From the pythagorean theorem we can calculate that y = x 5 so x = 2 5 .
The area of the blue shaded area is x 2 = 2 0 and the area of the orange shaded area is 1 0 2 − π 5 2 = 1 0 0 − 2 5 π .
The difference between them is 8 0 − 2 5 π = 5 ( 1 6 − 5 π ) . Therefore a = 5 and b = 1 6 and the digit sum of b is 7 ,
Finally B a a = 7 5 5 so the remainder is equal to 6 .