Question

Two circles intersect at points $A$ and $B.$ The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?

$2$

$1+\sqrt3$

$3$

$2+\sqrt3$

$4$

Answer: D

Concepts:chord trigonometry area ratio

Difficulty rating: 1630

Solution:

Let the circles have radii $R$ (with the $30^\circ$ arc) and $r$ (with the $60^\circ$ arc). The common chord has length $2R\sin15^\circ=2r\sin30^\circ,$ so $\dfrac{R}{r}=\dfrac{\sin30^\circ}{\sin15^\circ}.$

The smaller central angle gives the larger radius, so $R\gt r.$ The area ratio is $\begin{gathered} \left(\dfrac{R}{r}\right)^2\\ =\dfrac{1}{4\sin^2 15^\circ}\\ =\dfrac{1}{2(1-\cos30^\circ)}\\ =\dfrac{1}{2-\sqrt3}=2+\sqrt3. \end{gathered}$

Thus, the correct answer is D.

2014 AMC 12A Problem 12

Answer: D

Difficulty rating: 1630

Solution:

Problem 12 in Other Years