2007 AMC 12B Problems/Problem 20
Problem
The parallelogram bounded by the lines , , , and has area . The parallelogram bounded by the lines , , , and has area . Given that , , , and are positive integers, what is the smallest possible value of ?
Solution
Plotting the parallelogram on the coordinate plane, the 4 corners are at . Because , we have that or that , which gives (consider a homothety, or dilation, that carries the first parallelogram to the second parallelogram; because the area increases by , it follows that the stretch along the diagonal is ). The area of triangular half of the parallelogram on the right side of the y-axis is given by , so substituting :
Thus , and we verify that , will give us a minimum value for . Then .
See also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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All AMC 12 Problems and Solutions |