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

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New post 18 May 2016, 15:13
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A
B
C
D
E

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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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Re: If O is a center of a circle as above figure, what is the area of the  [#permalink]

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New post 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 8-)

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
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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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New post 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
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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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New post 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.
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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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New post 13 Oct 2017, 21:37
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