If in the figure above, AB is the diameter of a circle of radius 6, wh

Question

https://gmatclub.com/forum/download/file.php?id=52933 If in the figure above, AB is the diameter of a circle of radius 6, what is the perimeter of the shaded region? A. π+6+6√3 B. π+12+6√3 C. 2π+9+6√3 D. 2π+12+6√3 E. 2π+12+12√3 Are You Up For the Challenge: 700 Level Questions T6375.png

IanStewart · Answer

If you connect a diameter to a point on a circle, you always get a right angle at that point. So if we draw the triangle ABC here, it will have a 90 degree angle at C, and will therefore be a 30-60-90 triangle. It has a hypotenuse of length 12 (twice the radius), and since the sides of any 30-60-90 triangle are in a 1 to √3 to 2 ratio, the other two sides must be 6 and 6√3. The two longest of those sides, 6√3 and 12, are edges of the shaded region. So the answer is 12 + 6√3 + the length of the minor arc BC. So the answer is B or D. But the arc BC is clearly (slightly) longer than the straight line BC, which we just found was of length 6. From our remaining two answer choices, we can see that its length must be 2π, and not π, so the right answer must be D.

Of course you can also find the exactly length of that arc in other ways (for example, if O is the center of the triangle, it's easy to see that the triangle OCB will be equilateral, since two sides are radii and we found the third side CB above is also 6; that makes the angle COB a 60 degree angle, and minor arc BC is thus 60/360 = 1/6 of the circumference of the circle) but an estimate saves a bit of time.

Of course you can also find the exactly length of that arc in other ways (for example, if O is the center of the triangle, it's easy to see that the triangle OCB will be equilateral, since two sides are radii and we found the third side CB above is also 6; that makes the angle COB a 60 degree angle, and minor arc BC is thus 60/360 = 1/6 of the circumference of the circle) but an estimate saves a bit of time.

lacktutor · Answer

Please, FIND the attached file.

The answer is D.

globaldesi · Answer

the key here is angle A = angle C
and thus exterior angle is 60 
hence OCB is an equilateral triangle each side being 60 degree and length 6
and diameter AB = 12 cm
thus ACB is a 30 -60 - 90 triangle
hence sides are thus 6,  6\sqrt{3}  , 12
hence perimeter 
AB + AC + arc BC
=  12+ 6\sqrt{3}+\frac{ 60}{360} *2\pi 6 
 12 +6\sqrt{3} + 2\pi 
thus D

BrushMyQuant · Answer

Let's complete the figure as shown below: (Let O be the center of the circle and D be any point on the arc CB)

Attachment:

circle problem.png [ 19.2 KiB | Viewed 3029 times ]

Theory: Diameter subtends 90˚ at any point on the circle

=> ∠ACB = 90˚

Theory: Angle subtended by the chord at the center is twice the angle subtended by the chord at any other point on the circle.

=> ∠COB = 2*∠CAB = 2*30˚ = 60˚

And we know that OC = OB = Radius
=> ∠OCB = ∠OBC
=> ∠OCB + ∠OBC + 60˚ = 180˚
=> ∠OCB + ∠OCB = 120˚
=> ∠OCB = ∠OBC = \(\frac{120˚}{2}\) = 60˚
=> △ OCB = Equilateral Triangle => All sides = 6

Theory: Length of Arc which subtends angle Θ at the center = \(\frac{Θ}{360˚}∗2πr\)

=> Length of Arc CDB = \(\frac{60˚}{360˚}∗2π*6\)= 2π

Circumference of Shaded region = AB + CDB + AC = 2π + 12 + AC = 2π + 12 + AC

Length of AC

Using Pythagorean Theorem \(AC^2 = AB^2 - BC^2\) = \(12^2 - 6^2\) = 144-36 = 108 = 36*3 = \(6^2 * 3\)
=> AC = 6√3

=> Circumference of Shaded region = 2π + 12 + AC = 2π + 12 + 6√3

So, Answer will be D
Hope it helps!

Watch the following video to Learn Basics of Circles

If in the figure above, AB is the diameter of a circle of radius 6, wh

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If in the figure above, AB is the diameter of a circle of radius 6, wh

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