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:
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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.
The area of sector \(ABY=\frac{150}{360}*\pi{r^2}=\frac{5}{12}\pi{r^2}\);
The area of sector \(CBY=\frac{30}{360}*\pi{r^2}=\frac{1}{12}\pi{r^2}\);
The area of each of two small semicircles is \(\frac{\pi{(\frac{r}{2})^2}}{2}=\pi{\frac{r^2}{8}}\) (as its radius is half of the radius of the big circle);
The are of the shaded region above BY is \(\frac{5}{12}\pi{r^2}-\pi{\frac{r^2}{8}=\frac{7}{24}\pi{r^2}\);
The are of the shaded region below BY is \(\frac{1}{12}\pi{r^2}+\pi{\frac{r^2}{8}=\frac{5}{24}\pi{r^2}\);
Ratio of the areas of the shaded regions is \(\frac{7}{5}\).
Answer: D.
One query. As we know central angle of a circle is twice the inscribed angle. i.e. if inscribed angle is x then central angle is 2x.
So here if i see angle on YXA is 105 then my central angle on B should be 210.
Please clarify this.