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
Solution 1
Note as above that ABCD must be cyclic to obtain the circle with maximal radius. Let , , , and be the points on , , , and respectively where the circle is tangent. Let and . Since the quadrilateral is cyclic, and . Let the circle have center and radius . Note that , , , and are right angles.
Hence , , , and .
Therefore, and .
Let . Then , , , and . Using and we have , and . By equating the value of from each, . Solving we obtain so that .
Solution 2
To maximize the radius of the circle, we also need to maximize its area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. , so this particular quadrilateral has an incircle. By definition, given side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic.
For cyclic quadrilaterals, Brahmagupta's formula gives the area as where is the semiperimeter and and are the side lengths. Breaking it up into triangles, we see the area of a tangential quadrilateral is also equal to . Equate these two equations. Substituting , the semiperimeter, and , the area and solving for ,we get .
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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