2014 AMC 12A Problems/Problem 20
Contents
Problem
In , , , and . Points and lie on and respectively. What is the minimum possible value of ?
Solution 1
Let be the reflection of across , and let be the reflection of across . Then it is well-known that the quantity is minimized when it is equal to . (Proving this is a simple application of the triangle inequality; for an example of a simpler case, see Heron's Shortest Path Problem.) As lies on both and , we have . Furthermore, by the nature of the reflection, so . Therefore by the Law of Cosines
Solution 2
(Diagram by dasobson) Reflect across to . Similarly, reflect across to . Clearly, and . Thus, the sum . This value is maximized when , , and are collinear. To finish, we use the law of cosines on the triangle :
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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All AMC 12 Problems and Solutions |
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