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# Geometry Question

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In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\angle ADC$? Jun 28, 2018

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The area of the larger circle  will be  4 pi

So...the  shaded area  is  =  (5/12) * 4 pi  =  20/12 pi  =  ( 5/3 ) pi

Let  the measure  of  the smaller angle   ADC  = theta  (in rads)

And let the measure of the larger angle ADC  = [2pi - theta ]  in rads

So...the  area of the sector subtended by the large circle  is (1/2) 4^2 * θ =  2θ

And let the area of the sector subtended by the larger circle be  (1/2) * 1^2 * (2pi - θ) =

(1/2) (2pi - θ)

So we have that

2θ  +  (1/2)(2pi - θ)  =  (5/3)pi

2θ + pi - θ/2  =  (5/3)pi

(3/2)θ =  2/3 pi

θ = (2/3)(2/3) pi  =  (4/9) pi

So...the smaller angle ADC  =  (4/9)pi * 180 / pi  =   20 * 4   = 80°

Proof

Area of  shaded sector of lager  circle  = 2(4/9) pi  =  8/9 pi

Area of shaded  sector of smaller circle = (1/2) (2pi - (4/9) pi)  = (1/2) (14/9) pi = (7/9) pi

Combined areas =  (8/9 + 7/9) pi  =  (15/9) pi  =  (5/3) pi   Jun 29, 2018