2014 AMC 10B Problems/Problem 15
Contents
Problem
In rectangle , and points and lie on so that and trisect as shown. What is the ratio of the area of to the area of rectangle ?
Solution 1
Let the length of be , so that the length of is and .
Because is a rectangle, , and so . Thus is a right triangle; this implies that , so . Now drop the altitude from of , forming two triangles.
Because the length of is , from the properties of a triangle the length of is and the length of is thus . Thus the altitude of is , and its base is , so its area is .
To finish, .
Solution 2
WLOG, let and . Furthermore, drop an an altitude from to , which meets at . Since is right and has been trisected, it follows that and are both triangles. Therefore, , and . Hence, it follows that . Since is right, the height and base of are and , respectively. Thus, the area of is , and the area of rectengle is , so the ratio beween the area of and is . Note that we are able to assume that and because we were asked to find the ratio between two areas.
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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