2011 AMC 12A Problems/Problem 24
Problem
Consider all quadrilaterals such that , , , and . What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
Solution
Answer:
Given, a 14-9-7-12 quadrilateral ( which has an in-circle).
Find the largest possible in-radius.
Solution:
Since Area = semi-perimeter, and perimeter is fixed, we can maximize the area. Let the angle between the 14 and 12 be degree, and the one between the 9 and 7 be .
2(Area) =
(Area) =
By law of cosine,
(simple algebra left to the reader)
(Area) =
(Area) = , which reach maximum when .
(and since it is a quadrilateral, it is possible to have (hence cyclic quadrilateral, that would be the best guess and the extended Heron's formula which I forgot the name for would work for area and the work is simple).
(Area)
(Area)
(Area), Area = semi-perimeter.
Hence, , choice
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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All AMC 12 Problems and Solutions |