2013 AMC 12B Problems/Problem 20
Problem
For , points
and
are the vertices of a trapezoid. What is
?
$\textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D}}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Let be
(not respectively). Then we have four points
, and the two lines connecting two pairs of them must be parallel (as we are dealing with a trapezoid). WLOG, we take the line connecting the first two and the line connecting the last two to be parallel, so that
, or
.
Now, we must find how to match up to
so that the above equation has a solution. The three possible ways are
,
, and
. On the interval
, we have
, so the first two of those possible ways do not work because the LHS and the RHS have different ranges. Thus, it must be the last possible way,
.
Now, we perform some algebraic manipulation to find :
Solve the quadratic to find , so that
.
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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All AMC 12 Problems and Solutions |