2002 AMC 12B Problems/Problem 24
Contents
[hide]Problem
A convex quadrilateral with area contains a point in its interior such that . Find the perimeter of .
Solution 1
We have (This is true for any convex quadrilateral: split the quadrilateral along and then using the triangle area formula to evaluate and ), with equality only if . By the triangle inequality,
with equality if lies on and respectively. Thus
Since we have the equality case, at point , as shown below.
By the Pythagorean Theorem,
The perimeter of is .
Solution 2
Draw the diagram out. Notice the very peculiar sets of numbers ; these are simply multiples of the legs of well-known Pythagorean triples , pointing to signs of possible right angles. We can assume that , so the area of the entire figure would be:
Thus this is the correct case, so finding the side lengths of by the Pythagorean Theorem yields , ,,, so the perimeter is .
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See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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