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Manhattan GMAT challenge question

by gabriel » Fri Sep 21, 2007 5:56 am
Source : - Manhattan GMAT challenge question ( www.manhattangmat.com)

In the diagram to the right, points A, B, and C are on the diameter of the circle with center B. Additionally, all arcs pictured are semicircles. Suppose angle YXA = 105 degrees. What is the ratio of the area of the shaded region above the red line to the area of the shaded region below the red line? (Note: Diagram is not drawn to scale and angles drawn are not accurate.)

(A) 3/4
(B) 5/6
(C) 1
(D) 7/5
(E) 9/7

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by samirpandeyit62 » Fri Sep 21, 2007 7:51 am
given angle yxa =105 deg

join pts x and a with b

let angle xab =x , xyb =y

so angle angle yxb =y & axb =x (both are isoceles traingles)

so x+y =105

so in quadrilateral xyab we have

x+ y + x+ y + angle B =360

so angle B = 360 - 2(x+y)
=150 degs

now let diameter ABC = 8 so AB=4= diameter of smaller semicircle

so radius of smaller semicirle = 2

Now area of shaded region = 1/2 * pi * 16 - area of semicircle + area of semicircle

= 8pi

now area of shaded region above red line is

150/360 * pi*16 - area of semicirlce

= 20 pi /3 - 1/2*pi*4

=14pi/3

now area of shaded region below red line = 8pi - 14pi/3

=10 pi /3

so ratio is

14pi/3 /10pi/3

=7/5

so ans should be D
Regards
Samir
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by kajcha » Fri Sep 21, 2007 9:23 am
Suppose radius of the circle = x

angle YXA = 105. We know the property that angle formed by an arc at center is double of the angle formed ab the same arc on the opp side of the circle.

So, angle at center formed by ARC YCB = 105*2 = 210

angle YBC = 210-180 = 30
angle YBA = 180-30 = 150

area of the sector YBC = pi*x^2*30/360 = pi*x^2/12 -------(1)

area of the sector YBA = pi*x^2*150/360 = pi*x^2*5/12 -------(2)

area of the small semicircle (diameter x) = pi*(x/2)^2/2 = pi*x^2/8 -------(3)

area of the shaded region above red line = (2)-(3) = 7*pi*x^2/24 ------------(4)

area of the shaded regios below red line = (1)+(3) = 5*pi*x^2/24 --------(5)

ratio (4)/(5) = 7/5
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by gabriel » Sat Sep 22, 2007 11:35 pm
... Yup... 7/5 is my answer too ...
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