2011 AMC 10B Problems/Problem 20
Problem
Rhombus has side length and °. Region consists of all points inside the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?
Solution
Suppose that is a point in the rhombus and let be the perpendicular bisector of . Then if and only if is on the same side of as . The line divides the plane into two half-planes; let be the half-plane containing . Let us define similarly and . Then is equal to . The region turns out to be an irregular pentagon. We can make it easier to find the area of this region by dividing it into four triangles:
Since and are equilateral, contains , contains and , and contains . Then with and so and has area .
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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All AMC 10 Problems and Solutions |