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enigma123

In the diagram, 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 line YB to the area of the shaded region below the line YB? (Note: Diagram is not drawn to scale and angles drawn are not accurate.)

(A) ¾
(B) 5/6
(C) 1
(D) 7/5
(E) 9/7

Attachment:
Circle.png

Simple way using property of cyclic quadrilateral.
Make cyclic quadrilateral AXYC.Join AB-BC-YC
In AXYC:
AXY = 105
So ACY = 180-105 = 75 (Sum of opposite angles of cyclic quadrilateral is 180)

Make triangle YBC.
In YBC,YB = CB = radius
Hence BYC = BCY
But ACY = BCY (A-B-C)
So BYC = BCY = 75

YBC = 180-(BYC+BCY) = 180-150 = 30
So ABY = 150 (A-B-C is straight line)

Area of circular region AYB = 150/360 * Pi * r^2
Area of circular region BYC = 30/360 * Pi * r^2

Let radius be r,then area of semicircular region with radius as r/2 is (Pi*r^2)/8

Required ratio:

\(\frac{5}{12} *Pi*r^2 - Pi*r^2*\frac{ 1}{8}\)

Upon

\(\frac{1}{12} *Pi*r^2 + Pi*r^2*\frac{ 1}{8}\)

Which gives

\(\frac{7}{5}\)
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Bunuel
Genfi
Hi Bunel,

I didn't understand the below mentioned part. I did refer to the link provided by you. Can you please can explain this central angle theorem.

"According to the central angle theorem <ABY=2*(180-105)=150 (for more on this check Circles chapter of Math Book: math-circles-87957.html). Hence <CBY=180-150=30"

Thank you

Check the diagram below:
Attachment:
.png

The same as here:


Hope it's clear.


Hi Bunuel,
Why is the angle below the central angle alpha and not its (alpha's) adjacent one? Shouldn't the adjacent angle be alpha as it subtends the arc ?
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