2010 AMC 12B Problems/Problem 22
Problem 22
Let be a cyclic quadrilateral. The side lengths of
are distinct integers less than
such that
. What is the largest possible value of
?
Solution
Let ,
,
, and
. We see that by the Law of Cosines on
, we have
. Also,
. Now, we know that
. Also, because
is a cyclic quadrilateral, we must have that
, so
. Therefore,
. Now, adding, we have
.
We now look at the equation . Suppose that
. Then, we must have either
or
equal
. Suppose that
. We let
and
.
Now, , so it is
or
.
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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All AMC 12 Problems and Solutions |