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2013 UNCO Math Contest II Problems/Problem 3

Problem

[asy] filldraw(circle((0,-sqrt(2)/2),sqrt(2)/2),grey); filldraw(circle((0,0),1),white); draw((-sqrt(2)/2,-sqrt(2)/2)--(sqrt(2)/2,-sqrt(2)/2)--(0,0)--cycle,black); MP("C",(0,0),N);MP("A",(-sqrt(2)/2,-sqrt(2)/2),W);MP("B",(sqrt(2)/2,-sqrt(2)/2),E); [/asy]

Point $C$ is the center of a large circle that passes through both $A$ and $B$, and $C$ lies on the small circle whose diameter is $AB$. The area of the small circle is $9\pi$. Find the area of the shaded $\textit{lune}$, the region inside the small circle and outside the large circle.

Solution

From the information that \( 9\pi \) is the area of small circle, we can infer that \( \frac{1}{2} AB = 3 \). \(\because \text{area}_\text{small} = \pi (3)^{2} \implies \text{radius}_\text{small} = \frac{1}{2} AB = 3\)

The line from \( C \) to the center of the small circle, i.e., the midpoint of \( AB \), will be perpendicular to \( AB \).

Let the midpoint of \( AB \) be \( D \).

\[AD^{2} + CD^{2} = AC^{2}\] \[3^{2} + 3^{2} = AC^{2}\] \[AC^{2} = \text{radius}^{2}_\text{large} = 18\]

Now because \( A, B, C \) are part of the small triangle and \( AB \) is the diameter, the \( \angle ACB = 90^\circ \). Thus, the area of the sector \( ABC \) in the large circle:

\[\text{area}_{\text{sector}} = \frac{90}{360} \times \pi (\text{radius}_\text{large})^{2}\] \[= \frac{1}{4} \times 18 \pi\] \[= \boxed{ \frac{9}{2}\pi }\]

The area of the triangle \( ABC \):

\[\text{area}_{\triangle} = \frac{1}{2} \times CD \times AB\] \[= \frac{1}{2} \times 3 \times 6\] \[= 9\]

Area of the segment \( ABC \):

\[\text{area}_\text{segment} = \text{area}_\text{sector} - \text{area}_{\triangle}\] \[= \frac{9}{2} \pi - 9\] \[= 9\left( \frac{\pi}{2} -1 \right)\]

The area of the lune:

\[\text{area}_\text{lune} = \frac{1}{2} \text{area}_\text{small} - \text{area}_\text{segment}\] \[= \frac{1}{2} \cdot 9\pi - 9\left( \frac{\pi}{2} -1 \right)\] \[= 9 \left( \frac{\pi}{2} -\frac{\pi}{2} + 1 \right)\] \[= \boxed{ 9 }\]

See Also

2013 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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