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655-705 (Hard)|   Geometry|      
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MahmoudFawzy
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As shown in the figure above, Chord AB is drawn on a circle such that 28 Sep 2019, 14:46
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66% (03:02) correct 34% (03:12) wrong based on 173 sessions

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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) \(75π\)
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B
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As shown in the figure above, Chord AB is drawn on a circle such that 29 Sep 2019, 05:35
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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) \(75π\)
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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
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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 21:20
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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?
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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 22:11
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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?
Show more
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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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 22:27
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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?
Sum of angles in a triangle is 180° and 30,60,90 angles will be in the ratio 1:√3:2

Posted from my mobile device
Show more

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?
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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 22:56
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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?
Sum of angles in a triangle is 180° and 30,60,90 angles will be in the ratio 1:√3:2

Posted from my mobile device

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?
Show more
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
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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 23:16
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"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?
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As shown in the figure above, Chord AB is drawn on a circle such that 30 Sep 2019, 23:26
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chrtpmdr
"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?
Show more
it will not be mentioned in any question , it is a part of 30,60,90 property

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As shown in the figure above, Chord AB is drawn on a circle such that 19 Jul 2020, 19:55
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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.
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As shown in the figure above, Chord AB is drawn on a circle such that 02 Sep 2024, 02:47
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