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# If O is a center of a circle as above figure, what is the area of the

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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If O is a center of a circle as above figure, what is the area of the  [#permalink]

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18 May 2016, 15:13
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45% (medium)

Question Stats:

67% (02:28) correct 33% (02:17) wrong based on 92 sessions

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If O is a center of a circle as above figure, what is the area of the region shaded?
1) ∠AOB=30 degrees.
2) The length of arc CA is 2π (pi)

*An answer will be posted in 2 days.

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Joined: 23 Jul 2014 Posts: 91 Location: India WE: Information Technology (Computer Software) Re: If O is a center of a circle as above figure, what is the area of the [#permalink] ### Show Tags 20 May 2016, 19:32 MathRevolution wrote: If O is a center of a circle as above figure, what is the area of the region shaded? 1) ∠AOB=30 degrees. 2) The length of arc CA is 2π (pi) *An answer will be posted in 2 days. Let me be the first to Answer From what is given: ∠AOC =90 So, 1. area of the region AOC is 1/4 th of the total circle and 2. Circumference of AOC is 1/4th of the total circle From (1) ∠AOB=30 degrees. So ∠COB=120 degrees. This tells the area of region COB is 1/3rd of the total circle. But we do not get the value of none of the areas. Insufficient From(2) The length of arc CA is 2π from what is Given , we can tell that 2πr = 4.2π = 2π => r = 4 so we know the area of region AOC from this. But nothing else. This much information does not contribute to the process of finding So insufficient From (1) + (2) ∠COB=120 degrees. Area of region COB = 1/3 * π*16= 16π/3 Area of Triangle COB = 1/2*OC*OB*sin∠COB=1/2*r*r*sin120=1/2*4*4*cos30=4\sqrt{3} So area of region CAB = Area of region COB - Area of Triangle COB = 16π/3 -4\sqrt{3} Now Area of Triangle AOC = 1/2*4*4=8 Area of Region CA = area of region AOC - Now Area of Triangle AOC = 1/4* π*16 - 8 = 4π - 8 So Finally area of the Shaded region = area of region CAB - Area of Region CA = 16π/3 -4\sqrt{3} - 4π + 8 So Sufficient Answer C _________________ The Mind ~ Muscle Connection My GMAT Journey is Complete.Going to start the MBA in Information Management from 2016 Good Luck everyone. Current Student Status: Persevere Joined: 08 Jan 2016 Posts: 119 Location: Hong Kong GMAT 1: 750 Q50 V41 GPA: 3.52 Re: If O is a center of a circle as above figure, what is the area of the [#permalink] ### Show Tags 20 May 2016, 22:53 1 MathRevolution wrote: If O is a centre of a circle as above figure, what is the area of the region shaded? 1) ∠AOB=30 degrees. 2) The length of arc CA is 2π (pi) *An answer will be posted in 2 days. Area of shaded region = (Area of Triangle AOC + Area of minor Sector OAB) - Area of Triangle COB Triangle AOC is a right angled triangle at O and notice that OA and OC are radii of circle O. Let radius of circle O be 'r' So, Area of Triangle AOC = 1/2(OA)(OC) = r^2/2 Area of minor sector OAB = ∠AOB/360(πr^2) To find the Area of Triangle COB draw an altitude/ perpendicular OD from point O on base BC. Note that OC and OB are radii of Circle O. Hence, Triangle COB is an isosceles triangle. So the perpendicular OD drawn on base BC will bisect the BC and ∠BOC. Area of Triangle COB = 1/2(OD)(CB) Area of shaded region = r^2/2 + ∠AOB/360(πr^2) - 1/2(OD)(CB) So we need to know r, ∠AOB and length of OD and CB Statement 1 ∠AOB=30 degrees This statement does not provide any information about r, OD and CB. Hence, this is insufficient. We can eliminate options A and D Statement 2 The length of arc CA is 2π Since ∠AOC = 90, we know that length of arc CA = 1/4 Circumference of Circle O. We therefore, can find the value of r. 2π = 1/4(2πr) r = 4. But this statement does not provide us information about ∠AOB and length of OD and CB. Hence, this statement is insufficient. We can eliminate option B. Combining statement 1 and statement 2 We know r and ∠AOB. We need to find length of OD and CB Look at Isosceles triangle BOC. ∠COB = 120 (as ∠AOC = 90 and ∠AOB = 30). Hence, ∠CBO = ∠BCO = 30 Note that perpendicular bisector OD bisects the isosceles triangle BOC in two right angled triangles at D, i.e. Triangle BOD and Triangle COD. Both these triangles are 30-60-90 triangles. We can use the property of 30-60-90 triangles to find the length of OD, BD and DC. So, combining the statements gives us all the unknowns required to answer the question. Both statements combined are sufficient. We can eliminate option E. The correct answer is C. Hope this helps!! Nalin Attachments 1.png [ 32.6 KiB | Viewed 1439 times ] Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6517 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If O is a center of a circle as above figure, what is the area of the [#permalink] ### Show Tags 22 May 2016, 16:22 In the original condition, you need to know radius and the degree of AOB. Hence, there are 2 variables, which means C is likely the answer. In fact, If we use the condition 1) and the condition 2), C is the correct answer. - Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If O is a center of a circle as above figure, what is the area of the  [#permalink]

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13 Oct 2017, 21:37
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