Difference between revisions of "2019 AMC 12B Problems/Problem 25"
Numberwhiz (talk | contribs) (→Solution 3 (Complex Numbers)) |
Szhangmath (talk | contribs) (→Solution 5) |
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~r00tsOfUnity | ~r00tsOfUnity | ||
+ | |||
+ | ===Solution 5=== | ||
+ | |||
+ | |||
+ | Let <math>X, Y, Z</math> be the centroids of <math>\triangle ABC, \triangle BCD, \triangle ACD</math> respectively, then | ||
+ | |||
+ | <math>XZ//BD</math>, since <math>EX=\frac13 EB, EZ=\frac13 ED</math>, | ||
+ | |||
+ | <math>XY//GE</math>, since <math>BX=\frac23BE, BY=\frac23BG, EG//AD</math> by midsegment theorem, so <math>XY//AD</math> | ||
+ | |||
+ | Similarly, <math>YZ//AB</math>, | ||
+ | |||
+ | So <math>\triangle ABD</math> is an equilateral triangle | ||
+ | |||
+ | Assume <math>\alpha=\angle BCD</math>, then <math>BD^2=BC^2+CD^2-2\cdot BC\cdot CD\cos \alpha=40-24\cos\alpha</math>, the area | ||
+ | |||
+ | <math>[ABCD]=[ABD]+[BCD]=\frac{\sqrt3}4 BD^2+\frac12\cdot BC\cdot CD\sin\alpha=</math> | ||
+ | |||
+ | <math>10\sqrt3+6(\sin\alpha-\sqrt3\cos \alpha)=10\sqrt3+12\sin(\alpha-60^\circ)</math> | ||
+ | |||
+ | |||
+ | The maximal value happens when <math>\sin(\alpha-60^\circ)=1</math>, and the value is <math>10\sqrt3+12</math>, and the answer is <math>\boxed{\textbf{(C)} 12+10\sqrt3}</math>. | ||
+ | |||
+ | <asy> | ||
+ | import graph; size(11.42cm); | ||
+ | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||
+ | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ | ||
+ | pen dotstyle = black; /* point style */ | ||
+ | real xmin = -1.58, xmax = 9.84, ymin = -7.74, ymax = 8.48; /* image dimensions */ | ||
+ | |||
+ | /* draw figures */ | ||
+ | draw((0.,0.)--(4.,0.), linewidth(2.)); | ||
+ | draw((2.,3.4641016151377544)--(4.,0.), linewidth(2.)); | ||
+ | draw((0.,0.)--(2.,3.4641016151377544), linewidth(2.)); | ||
+ | draw((4.,0.)--(5.,1.), linewidth(2.)); | ||
+ | draw((5.,1.)--(2.,3.4641016151377544), linewidth(2.)); | ||
+ | draw((0.,0.)--(5.,1.), linewidth(2.)); | ||
+ | draw((2.5,0.5)--(4.,0.), linewidth(2.)); | ||
+ | draw((4.,0.)--(3.5,2.232050807568877), linewidth(2.)); | ||
+ | draw((2.,3.4641016151377544)--(2.5,0.5), linewidth(2.)); | ||
+ | draw((0.,0.)--(3.5,2.232050807568877), linewidth(2.)); | ||
+ | draw((0.,0.)--(4.5,0.5), linewidth(2.)); | ||
+ | draw((2.,3.4641016151377544)--(4.5,0.5), linewidth(2.)); | ||
+ | draw((2.333333333333333,1.4880338717125845)--(3.,0.3333333333333333), linewidth(2.) + linetype("2 2")); | ||
+ | draw((3.666666666666666,1.488033871712585)--(3.,0.3333333333333333), linewidth(2.) + linetype("2 2")); | ||
+ | /* dots and labels */ | ||
+ | dot((0.,0.),dotstyle); | ||
+ | label("$A$", (-0.2,-0.18), NE * labelscalefactor); | ||
+ | dot((4.,0.),dotstyle); | ||
+ | label("$B$", (3.98,-0.3), NE * labelscalefactor); | ||
+ | dot((2.,3.4641016151377544),dotstyle); | ||
+ | label("$D$", (1.86,3.62), NE * labelscalefactor); | ||
+ | dot((5.,1.),dotstyle); | ||
+ | label("$C$", (5.06,0.92), NE * labelscalefactor); | ||
+ | dot((2.5,0.5),linewidth(4.pt) + dotstyle); | ||
+ | label("$E$", (2.2,0.5), NE * labelscalefactor); | ||
+ | dot((4.5,0.5),linewidth(4.pt) + dotstyle); | ||
+ | label("$F$", (4.52,0.3), NE * labelscalefactor); | ||
+ | dot((3.5,2.232050807568877),linewidth(4.pt) + dotstyle); | ||
+ | label("$G$", (3.58,2.4), NE * labelscalefactor); | ||
+ | dot((2.333333333333333,1.4880338717125845),linewidth(4.pt) + dotstyle); | ||
+ | label("$Z$", (2.06,1.44), NE * labelscalefactor); | ||
+ | dot((3.,0.3333333333333333),linewidth(4.pt) + dotstyle); | ||
+ | label("$X$", (2.92,0.04), NE * labelscalefactor); | ||
+ | dot((3.666666666666666,1.488033871712585),linewidth(4.pt) + dotstyle); | ||
+ | label("$Y$", (3.76,1.5), NE * labelscalefactor); | ||
+ | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); | ||
+ | </asy> | ||
+ | |||
+ | |||
+ | ~szhangmath | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2019|ab=B|num-b=24|after=Last Problem}} | {{AMC12 box|year=2019|ab=B|num-b=24|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:37, 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
Let be the centroids of
respectively, then
, since
,
, since
by midsegment theorem, so
![]()
Similarly,,
Sois an equilateral triangle
Assume, then
, the area
The maximal value happens when, and the value is
, and the answer is
.
~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.