Difference between revisions of "2018 AMC 12B Problems/Problem 13"
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</asy> | </asy> | ||
+ | <math>\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}</math> | ||
+ | |||
+ | ==Solution 1 (Similar Triangles)== | ||
+ | As shown below, let <math>M_1,M_2,M_3,M_4</math> be the midpoints of <math>\overline{AB},\overline{BC},\overline{CD},\overline{DA},</math> respectively, and <math>G_1,G_2,G_3,G_4</math> be the centroids of <math>\triangle{ABP},\triangle{BCP},\triangle{CDP},\triangle{DAP},</math> respectively. | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | unitsize(210); | ||
+ | pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); | ||
+ | pair M1 = midpoint(A--B); | ||
+ | pair M2 = midpoint(B--C); | ||
+ | pair M3 = midpoint(C--D); | ||
+ | pair M4 = midpoint(D--A); | ||
+ | pair G1 = centroid(A,B,P); | ||
+ | pair G2 = centroid(B,C,P); | ||
+ | pair G3 = centroid(C,D,P); | ||
+ | pair G4 = centroid(D,A,P); | ||
+ | filldraw(M1--M2--P--cycle,red); | ||
+ | filldraw(M2--M3--P--cycle,yellow); | ||
+ | filldraw(M3--M4--P--cycle,green); | ||
+ | filldraw(M4--M1--P--cycle,lightblue); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(M1--M2--M3--M4--cycle); | ||
+ | draw(G1--G2--G3--G4--cycle); | ||
+ | dot(P); | ||
+ | defaultpen(fontsize(10pt)); | ||
+ | draw(A--P--B); | ||
+ | draw(C--P--D); | ||
+ | label("$A$", A, W); | ||
+ | label("$B$", B, W); | ||
+ | label("$C$", C, E); | ||
+ | label("$D$", D, E); | ||
+ | label("$P$", P, N); | ||
+ | label("$M_1$", M1, W); | ||
+ | label("$M_2$", M2, S); | ||
+ | label("$M_3$", M3, E); | ||
+ | label("$M_4$", M4, N); | ||
+ | label("$G_1$", G1, 1.5S); | ||
+ | label("$G_2$", G2, 1.5E); | ||
+ | label("$G_3$", G3, 1.5NE); | ||
+ | label("$G_4$", G4, 1.5E); | ||
+ | dot(A); | ||
+ | dot(B); | ||
+ | dot(C); | ||
+ | dot(D); | ||
+ | dot(M1); | ||
+ | dot(M2); | ||
+ | dot(M3); | ||
+ | dot(M4); | ||
+ | dot(G1); | ||
+ | dot(G2); | ||
+ | dot(G3); | ||
+ | dot(G4); | ||
+ | </asy> | ||
+ | By SAS, we conclude that <math>\triangle G_1G_2P\sim\triangle M_1M_2P, \triangle G_2G_3P\sim\triangle M_2M_3P, \triangle G_3G_4P\sim\triangle M_3M_4P,</math> and <math>\triangle G_4G_1P\sim\triangle M_4M_1P.</math> By the properties of centroids, the ratio of similitude for each pair of triangles is <math>\frac{2}{3}.</math> | ||
+ | |||
+ | Note that quadrilateral <math>M_1M_2M_3M_4</math> is a square of side-length <math>15\sqrt2.</math> It follows that: | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li>Since <math>\overline{G_1G_2}\parallel\overline{M_1M_2},\overline{G_2G_3}\parallel\overline{M_2M_3},\overline{G_3G_4}\parallel\overline{M_3M_4},</math> and <math>\overline{G_4G_1}\parallel\overline{M_4M_1}</math> by the Converse of the Corresponding Angles Postulate, we have <math>\angle G_1G_2G_3=\angle G_2G_3G_4=\angle G_3G_4G_1=\angle G_4G_1G_2=90^\circ.</math></li><p> | ||
+ | <li>Since <math>G_1G_2=\frac23M_1M_2, G_2G_3=\frac23M_2M_3, G_3G_4=\frac23M_3M_4,</math> and <math>G_4G_1=\frac23M_4M_1</math> by the ratio of similitude, we have <math>G_1G_2=G_2G_3=G_3G_4=G_4G_1=10\sqrt2.</math></li><p> | ||
+ | </ol> | ||
+ | Together, quadrilateral <math>G_1G_2G_3G_4</math> is a square of side-length <math>10\sqrt2,</math> so its area is <math>\left(10\sqrt2\right)^2=\boxed{\textbf{(C) }200}.</math> | ||
+ | |||
+ | <u><b>Remark</b></u> | ||
+ | |||
+ | This solution shows that, if point <math>P</math> is within square <math>ABCD,</math> then the shape and the area of quadrilateral <math>G_1G_2G_3G_4</math> are independent of the location of <math>P.</math> Let the brackets denote areas. More generally, <math>G_1G_2G_3G_4</math> is always a square of area <cmath>[G_1G_2G_3G_4]=\left(\frac23\right)^2[M_1M_2M_3M_4]=\frac49[M_1M_2M_3M_4]=\frac29[ABCD].</cmath> On the other hand, the location of <math>G_1G_2G_3G_4</math> is dependent on the location of <math>P.</math> | ||
+ | |||
+ | ~RandomPieKevin ~Kyriegon ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2 (Similar Triangles)== | ||
+ | This solution refers to the diagram in Solution 1. | ||
+ | |||
+ | By SAS, we conclude that <math>\triangle G_1G_3P\sim\triangle M_1M_3P</math> and <math>\triangle G_2G_4P\sim\triangle M_2M_4P.</math> By the properties of centroids, the ratio of similitude for each pair of triangles is <math>\frac23.</math> | ||
+ | |||
+ | Note that quadrilateral <math>M_1M_2M_3M_4</math> is a square of diagonal-length <math>30,</math> so <math>\overline{M_1M_3}\perp\overline{M_2M_4}.</math> Since <math>\overline{G_1G_3}\parallel\overline{M_1M_3}</math> and <math>\overline{G_2G_4}\parallel\overline{M_2M_4}</math> by the Converse of the Corresponding Angles Postulate, we have <math>\overline{G_1G_3}\perp\overline{G_2G_4}.</math> | ||
+ | |||
+ | Therefore, the area of quadrilateral <math>G_1G_2G_3G_4</math> is <cmath>\frac12\cdot G_1G_3\cdot G_2G_4 = \frac12\cdot\left(\frac23\cdot M_1M_3\right)\cdot\left(\frac23\cdot M_2M_4\right)=\boxed{\textbf{(C) }200}.</cmath> | ||
+ | ~Funnybunny5246 ~MRENTHUSIASM | ||
− | <math> | + | ==Solution 3 (Coordinate Geometry)== |
+ | This solution refers to the diagram in Solution 1. | ||
+ | |||
+ | We place the diagram in the coordinate plane: Let <math>A=(0,30),B=(0,0),C=(30,0),D=(30,30),</math> and <math>P=(3x,3y).</math> | ||
+ | |||
+ | Recall that for any triangle in the coordinate plane, the coordinates of its centroid are the averages of the coordinates of its vertices. It follows that <math>G_1=(x,y+10),G_2=(x+10,y),G_3=(x+20,y+10),</math> and <math>G_4=(x+10,y+20).</math> | ||
+ | |||
+ | Note that <math>G_1G_3=G_2G_4=20</math> and <math>\overline{G_1G_3}\perp\overline{G_2G_4}.</math> Therefore, the area of quadrilateral <math>G_1G_2G_3G_4</math> is <cmath>\frac12\cdot G_1G_3\cdot G_2G_4=\boxed{\textbf{(C) }200}.</cmath> | ||
+ | |||
+ | ~Pi31415926535897 ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 4 (Homothety)== | ||
+ | |||
+ | Let <math>X,Y,Z,W</math> be the midpoints of sides <math>AB,BC,CD,DE</math>, respectively. | ||
− | + | Notice that a homothety centered at P with ratio <math>\frac{2}{3}</math> will send <math>XYZW</math> to <math>G_{1}G_{2}G_{3}G_{4}</math>, so <math>G_{1}G_{2}G_{3}G_{4}</math> is a square with area <math>\left(\frac{2}{3}\right)^2 [XYZW]</math>, but <math>[XYZW]=\frac{1}{2}[ABCD]</math> so our desired area is <cmath>\frac{4}{9}\cdot\frac{1}{2}\cdot900=\boxed{\textbf{(C) }200}</cmath> | |
− | |||
− | + | ~chrisdiamond10 | |
− | ==Solution | + | == Video Solution (Meta-Solving Technique) == |
− | + | https://youtu.be/GmUWIXXf_uk?t=1439 | |
− | + | ~ pi_is_3.14 | |
− | |||
==See Also== | ==See Also== | ||
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{{AMC12 box|year=2018|ab=B|num-b=12|num-a=14}} | {{AMC12 box|year=2018|ab=B|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 06:53, 31 August 2024
Contents
Problem
Square has side length
. Point
lies inside the square so that
and
. The centroids of
,
,
, and
are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
Solution 1 (Similar Triangles)
As shown below, let be the midpoints of
respectively, and
be the centroids of
respectively.
By SAS, we conclude that
and
By the properties of centroids, the ratio of similitude for each pair of triangles is
Note that quadrilateral is a square of side-length
It follows that:
- Since
and
by the Converse of the Corresponding Angles Postulate, we have
- Since
and
by the ratio of similitude, we have
Together, quadrilateral is a square of side-length
so its area is
Remark
This solution shows that, if point is within square
then the shape and the area of quadrilateral
are independent of the location of
Let the brackets denote areas. More generally,
is always a square of area
On the other hand, the location of
is dependent on the location of
~RandomPieKevin ~Kyriegon ~MRENTHUSIASM
Solution 2 (Similar Triangles)
This solution refers to the diagram in Solution 1.
By SAS, we conclude that and
By the properties of centroids, the ratio of similitude for each pair of triangles is
Note that quadrilateral is a square of diagonal-length
so
Since
and
by the Converse of the Corresponding Angles Postulate, we have
Therefore, the area of quadrilateral is
~Funnybunny5246 ~MRENTHUSIASM
Solution 3 (Coordinate Geometry)
This solution refers to the diagram in Solution 1.
We place the diagram in the coordinate plane: Let and
Recall that for any triangle in the coordinate plane, the coordinates of its centroid are the averages of the coordinates of its vertices. It follows that and
Note that and
Therefore, the area of quadrilateral
is
~Pi31415926535897 ~MRENTHUSIASM
Solution 4 (Homothety)
Let be the midpoints of sides
, respectively.
Notice that a homothety centered at P with ratio will send
to
, so
is a square with area
, but
so our desired area is
~chrisdiamond10
Video Solution (Meta-Solving Technique)
https://youtu.be/GmUWIXXf_uk?t=1439
~ pi_is_3.14
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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.