Challenge problem question -

[nsr2000vz]

This is from the MGMAT challenge problems:

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

Post Date: 09/17/07 (2007 Jul-Sept - Yin and Yang) (There is a picture Circle with Yin/Yang )

I have gone through the solution but I don't understand:

The solution given:

Inscribed angle AXY (given as 105°) intercepts arc ACY. By definition, the measure of an inscribed angle is equal to half the measure of its intercepted arc. Thus arc ACY = 2 x 105 = 210°.

I don't understand how AXY intercepts ACY?

Please explain,
Thanks,
Shivan

[nsr2000vz]

Attached, image!

[/img]

[RonPurewal]

I don't understand how AXY intercepts ACY?

perhaps you're thinking of the word "intersects"?

the intercepted arc is the arc that's "cut off" by an angle; it's the part of the circle that lies in the interior of the angle.

here's another way of thinking about it:
imagine that the angle is an "alligator mouth"; i.e., imagine that there are alligator teeth on the inside of the angle's sides.
in this image, the intercepted arc is the arc that's being "eaten" by the angle.

a nice animation is here.

[sprparvathy]

Hi, I'm still unable to understand the solution to this problem. Would be glad if someone gives a detailed step-by-step solution. Thank you,

[ps63739]

If AXY is 105 degrees, there could be a lot of combinations of the red line. And so the ratios of ares.
Unless AX is tangent to circle. Then point A and X will represent the same point. Then BAY cannot be 105 degree.
Is that the all information provided in the question?

[tim]

Okay the most straightforward way to do this one is to calculate the two areas directly. Start with the information that AXY is 105 degrees. then arc ACY is 210 degrees and arc AXY is 150 degrees. We will need this information to determine how much of the circle we are dealing with in each case. We will also assign a diameter of 4 to the large circle for convenience. Notice that there is no information that will help us determine this radius, and the problem asks for a ratio; in a case like this it is permissible (and advisable) to assign a value that works..

Below the line, let's split the region into two parts: a semicircle and a sector (pie piece). The semicircle has radius 1 so its area is pi/2. Because arc ACY is 210, arc CY is 30 degrees, which means the sector takes up 1/12 of the big circle. As the big circle has area 4pi, the sector has area pi/3. This gives us 5/6*pi for the area below the line..

Above the line, the region is a sector with a semicircle cut out. Because arc AXY is 150 degrees, the sector is 150/360 of a whole circle. 150/360*4pi = 5/3*pi. Take away the semicircle of area pi/2 and we get 7/6*pi above the line..

Taking the ratio of these two areas (the pi and 6 drop out), we get 7/5..