Problem:
Suppose , and are points in the plane with and , and let be the length of the line segment from to the midpoint of . Define a function by letting be the area of . Then the domain of is an open interval , and the maximum value of occurs at . What is ?
Answer Choices:
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Solution:
By the Triangle Inequality, is between and . The corresponding bounding values of are and .\
The area of is maximized when is a right angle, in which case the area is , and the median to the hypotenuse has half the length of the hypotenuse, namely
(This right triangle is the double of the right triangle.) The requested sum is .
Note: To compute the area of with and given, as a function of the length of the median to , extend this median a distance beyond to a point . Because diagonals and bisect each other, is a parallelogram, and its area is twice the area of and also twice the area of , whose side lengths are , and . This value can be computed by Heron's formula.
The problems on this page are the property of the MAA's American Mathematics Competitions