Difference between revisions of "2019 AMC 12B Problems/Problem 25"
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===Solution 5=== | ===Solution 5=== | ||
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Revision as of 12:58, 29 August 2024
Contents
Problem
Let be a convex quadrilateral with
and
Suppose that the centroids of
and
form the vertices of an equilateral triangle. What is the maximum possible value of the area of
?
Solution 1 (vectors)
Place an origin at , and assign position vectors of
and
. Since
is not parallel to
, vectors
and
are linearly independent, so we can write
for some constants
and
. Now, recall that the centroid of a triangle
has position vector
.
Thus the centroid of is
; the centroid of
is
; and the centroid of
is
.
Hence ,
, and
. For
to be equilateral, we need
. Further,
. Hence we have
, so
is equilateral.
Now let the side length of be
, and let
. By the Law of Cosines in
, we have
. Since
is equilateral, its area is
, while the area of
is
. Thus the total area of
is
, where in the last step we used the subtraction formula for
. Alternatively, we can use calculus to find the local maximum. Observe that
has maximum value
when e.g.
, which is a valid configuration, so the maximum area is
.
Solution 2
Let ,
,
be the centroids of
,
, and
respectively, and let
be the midpoint of
.
,
, and
are collinear due to well-known properties of the centroid. Likewise,
,
, and
are collinear as well. Because (as is also well-known)
and
, we have
. This implies that
is parallel to
, and in terms of lengths,
. (SAS Similarity)
We can apply the same argument to the pair of triangles and
, concluding that
is parallel to
and
. Because
(due to the triangle being equilateral),
, and the pair of parallel lines preserve the
angle, meaning
. Therefore
is equilateral.
At this point, we can finish as in Solution 1, or, to avoid using trigonometry, we can continue as follows:
Let , where
due to the Triangle Inequality in
. By breaking the quadrilateral into
and
, we can create an expression for the area of
. We use the formula for the area of an equilateral triangle given its side length to find the area of
and Heron's formula to find the area of
.
After simplifying,
Substituting , the expression becomes
We can ignore the for now and focus on
.
By the Cauchy-Schwarz inequality,
The RHS simplifies to , meaning the maximum value of
is
.
Thus the maximum possible area of is
.
Solution 3 (Complex Numbers)
Let ,
,
, and
correspond to the complex numbers
,
,
, and
, respectively. Then, the complex representations of the centroids are
,
, and
. The pairwise distances between the centroids are
,
, and
, all equal. Thus,
, so
. Hence,
is equilateral.
By the Law of Cosines,
.
. Thus, the maximum possible area of
is
.
~ Leo.Euler
Solution 4 (Homothety)
Let , and
be the centroids of
, and
, respectively, and let
and
be the midpoints of
and
, respectively. Note that
and
are
of the way from
to
and
, respectively, by a well-known property of centroids. Then a homothety centered at
with ratio
maps
and
to
and
, respectively, implying that
is equilateral too. But
is the medial triangle of
, so
is also equilateral. We may finish with the methods in the solutions above.
~ numberwhiz
Video Solution by MOP 2024
~r00tsOfUnity
Solution 5
\usepackage{enumerate}
\begin{enumerate}[(i.)]
\item
Let be the centroids of
respectively, then
\item
, since
,
\item
, since
by midsegment theorem, so
\item Similarly,
,
\item So
is an equilateral triangle
\item Assume
, then
, the area
\item
\item The maximal value happens when , and the value is
, (C)
\end{enumerate}
~szhangmath
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.