2010 AIME II Problems/Problem 9
Let be a regular hexagon. Let , , , , , and be the midpoints of sides , , , , , and , respectively. The segments , , , , , and bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of be expressed as a fraction where and are relatively prime positive integers. Find .
Let be the intersection of and
and be the intersection of and .
Let be the center.
Let (without loss of generality).
Note that is the vertical angle to an angle of regular hexagon, and so has degree .
Because and are rotational images of one another, we get that and hence .
Using a similar argument, , and
Applying the Law of cosines on ,
Thus, the answer is 4 + 7 = .
We can use coordinates. Let be at with at ,
then is at ,
is at ,
is at ,
Line has the slope of and the equation of
Line has the slope of and the equation
Let's solve the system of equation to find
Thus, the answer is .
|2010 AIME II (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|