As shown in the figure above, Chord AB is drawn on a circle such that

Question

Untitled.png As shown in the figure above, Chord AB is drawn on a circle such that the shortest distence from the chord to the center of the circle, C , is half the radius. If the diameter of the circle is 20, what is the area of the shaded portion of the figure? (A) \frac{100π}{3} - \frac{25\sqrt{3}}{2} (B) \frac{100π}{3} - 25\sqrt{3} (C) 50π - 5\sqrt{3} (D) 125 (E) 75π

chetan2u · Answer

Step I -  Join center with the end points of chord AB..
 Step II -  Take  triangle BCD ... BC = 10 and CD is half of CD, so triangle has sides  1:\sqrt{3}:2 , so angles are 30-60-90.
 Step III -   angle BCD = 60   as it is opposite  \sqrt{3} , so angle ACB will be TWICE =  60*2=120 .
 Step IV   Area of sector ACB  = 120/360=1/3 of Total area of circle =  \frac{1}{3}*\pi*10^2=\frac{100\pi}{3} 
 Step V -    Area of triangle ACB  =  \frac{1}{2}*AB*CD=\frac{1}{2}*2*5\sqrt{3}*5=25\sqrt{3}

Now Area of shaded portion = Area of sector ACB-area of triangle ACB =  \frac{100\pi}{3}-25\sqrt{3}

B

chrtpmdr · Answer

Why can we assume that the angle is 60 and therefore it is a 30/60/90?
I had the same draft but I did not know this was a property - I knew it must be the case when I saw the answer choices though.

Is this a certain property?

Peddi · Answer

Sum of angles in a triangle is 180° and 30,60,90 angles will be in the ratio 1:√3:2

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chrtpmdr · Answer

I know the 30/60/90 properties; I just don't understand why we are allowed to assume that it is one in this case. 
We are given a 90 Degree angle only, couldn't it be the case that one angle is 61 degree and the other one 29 degree which therefore would not allow to use the 30/60/90 rules?

Peddi · Answer

The ponint here is the converse is also true if the sides are in ratio 1:√3:2 then the angles are 30,60,90 . the angles you suggested doesnt give the sides ratio 1:√3:2

chrtpmdr · Answer

"The ponint here is the converse is also true if the sides are in ratio 1:√3:2"

Where in the problem text is that given?

Peddi · Answer

it will not be mentioned in any question , it is a part of 30,60,90 property

Posted from my mobile device

Logo · Answer

Super late here and I am sure you are not worried about this anymore. But to address your concern of the 60 degree angle - we know it's 60 degrees because the length of b-c is 10 (since it is a line coming from the center of the circle to the perimeter), therefore with the given length of 5 and the length of 10 you will see a 30-60-90 degree triangle.

bumpbot · Answer

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As shown in the figure above, Chord AB is drawn on a circle such that

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As shown in the figure above, Chord AB is drawn on a circle such that

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