2021 Fall AMC 12A Problems/Problem 24
Problem
Convex quadrilateral has
, and
. In some order, the lengths of the four sides form an arithmetic progression, and side
is a side of maximum length. The length of another side is
. What is the sum of all possible values of
?
Solution
Let be a point on
such that
is a parallelogram. Suppose that
and
so
as shown below.
DIAGRAM WILL BE READY VERY VERY SOON ...
We apply the Law of Cosines to
Let
be the common difference of the arithmetic progression of the side-lengths. It follows that
and
are
and
in some order. Note that
If then
is a rhombus with side-length
which is valid.
If then we have six cases:
WILL COMPLETE VERY SOON. A MILLION THANKS FOR NOT EDITING THIS PAGE.
~MRENTHUSIASM
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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