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805+ Level|   Geometry|      
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MrJglass
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In the diagram, points A, B, and C are on the diameter of 17 Jul 2018, 17:32
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Bunuel


Hope it's clear.
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I solved this question in less than 10 seconds, I was surprised to find out that it is 95% hard. I probably did something wrong and was lucky to get the correct answer. Please look at what I did and correct me, where I'm wrong.

Since the question stem indicates that the diagram is not drawn to scale, using the provided info, I drew a proper diagram with decent semi-circles.

Instead of using angle YXA as 105°, I turned it upside down and used angle ABY as 105°. Since we are only looking for the ratio of the regions, irrespective of which position the degree is coming from.

Thus,I had the upper shaded region to be 105°, and the lower region to be therefore (360-(180+105)) = 75°.

I went ahead to compute 105/75 = 7/5.

It might seem pretty long but it was pretty fast. Although it seems like a ludicrous luck!
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In the diagram, points A, B, and C are on the diameter of 23 Aug 2020, 19:19
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without the addition of a picture,


when an Inscribed Angle like Angle YXA touches the circumference of the circle on the SAME SIDE of the Circle that it is Inscribed upon, then I always like to think of the "Central Angle" as the Reflex Angle.

What I mean is if you draw a Radius from B to A, then you have Angle YBA "Touching" the same 2 Points (Point A and Point Y) as Inscribed Angle YXA does.

The Central Angle that measures the Major Arc from A all the way around the circle to C and then Y will be measured by the Reflex Angle on the Opposite Side of Angle YBA. That Reflex Angle will Equal 2 * (Inscribed Angle of 105 degrees).

If that Reflex Angle = 210 degrees, then the Angle YBA will = 150 degrees because together they are 360 degrees around Point B-Center.

I don't know if this helped anyone. I'll try to draw a picture and get it attached if someone is interested.
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In the diagram, points A, B, and C are on the diameter of 02 Jan 2021, 04:51
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arichinna
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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It can also be solved the following way:

Three new triangles can be formed which will be isosceles triangles (please refer the attachment)
Hence,
x+y = 105 (given)
2(x+y+z) = 360 since ACYX is a quadrilateral
x+y+z = 180
-> z = 75

in Triangle, BYC, <YBC will be 180 - 2*z = 180 - 2*75 = 30 degrees.

(1) Area of the total shaded portion is half the area of the circle pi*(r^2)/2

(2) Area below the red line = area of segment BYC of circle + area of shaded semicircle BC
= (30/360)*(pi*r^2) + pi ((r/2)^2)/2
=pi*(r^2)/12+pi*(r^2)/8
=5 * pi * (r^2) / 24

(3) Area above the red line is (1)-(2) above
= [ pi * (r^2)/2 ] - [ 5 * pi* (r^2) / 24 ]
= 7 * pi * (r^2) / 24

Answer is (3) / (2) which is 7/5

Kudos if you like the post :)
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Hello, could you please explain further how the area above the line is 1-2
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In the diagram, points A, B, and C are on the diameter of 16 Aug 2021, 13:32
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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

Show Spoiler
Attachment:
Circle.png
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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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In the diagram, points A, B, and C are on the diameter of 13 Oct 2022, 07:25
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Bunuel
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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.
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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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In the diagram, points A, B, and C are on the diameter of 20 Oct 2023, 05:20
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