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
Template:Incomplete
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
:
\[\frac{1}{2} (c-d)\left(\frac{c-d}{a-b}\right) &= 9 \quad \Longrightarrow \quad 2d^2 &= 9(a-b)\] (Error compiling LaTeX. Unknown error_msg)
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 |