Question

Two concentric circles are centered at point p. the sides of a 45 degree angle at p form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at p. what is the ratio of the area of the smaller circle to the area of the larger circle? express your answer as a common fraction.

Answer 1

To find the ratio of the areas of two concentric circles, we set up a proportion using their radii. By comparing the lengths of the arcs formed by the angles at the circles' center, we can find the ratio of the radii. The ratio of the area of the smaller circle to the area of the larger circle is 25/16.

Step-by-step explanation:

To find the ratio of the area of the smaller circle to the area of the larger circle, we need to compare their radii. Let's assume the radius of the smaller circle is 'r' and the radius of the larger circle is 'R'.

Since the lengths of the arcs are equal, we can set up the following proportion: angle in the smaller circle/angle in the larger circle = arc length in the smaller circle/arc length in the larger circle. That gives us: 45 degrees / 36 degrees = 2πr / 2πR.

Simplifying the equation, we get 45/36 = r/R. Cross multiplying, 45R = 36r. Finally, by dividing both sides by R, we get r/R = 45/36 = 5/4.

Therefore, the ratio of the area of the smaller circle to the area of the larger circle is (r^2)/(R^2) = (5/4)^2 = 25/16.

Answer 2

Together with triangles, circles comprise most of the GMAT Geometry problems.

A circle is the set of all points on a plane at the same distance from a single point ("the center").

The boundary line of a circle is called the circumference.