Difference between revisions of "2018 AMC 12B Problems/Problem 13"

Revision as of 17:28, 24 September 2021

Problem

Square $ABCD$ has side length $30$ . Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$ . The centroids of $\triangle{ABP}$ , $\triangle{BCP}$ , $\triangle{CDP}$ , and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

$[asy] unitsize(120); pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); draw(A--B--C--D--cycle); dot(P); defaultpen(fontsize(10pt)); draw(A--P--B); draw(C--P--D); label("$A$", A, W); label("$B$", B, W); label("$C$", C, E); label("$D$", D, E); label("$P$", P, N*1.5+E*0.5); dot(A); dot(B); dot(C); dot(D); [/asy]$

$\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}$

Solution 1 (Similar Triangles and Area Ratios)

As shown below, let $M_1,M_2,M_3,M_4$ be the midpoints of $\overline{AB},\overline{BC},\overline{CD},\overline{DA},$ respectively, and $G_1,G_2,G_3,G_4$ be the centroids of $\triangle{ABP},\triangle{BCP},\triangle{CDP},\triangle{DAP},$ respectively. $[asy] unitsize(210); pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3); pair M1 = midpoint(A--B); pair M2 = midpoint(B--C); pair M3 = midpoint(C--D); pair M4 = midpoint(D--A); pair G1 = centroid(A,B,P); pair G2 = centroid(B,C,P); pair G3 = centroid(C,D,P); pair G4 = centroid(D,A,P); filldraw(M1--M2--P--cycle,red); filldraw(M2--M3--P--cycle,yellow); filldraw(M3--M4--P--cycle,green); filldraw(M4--M1--P--cycle,lightblue); draw(A--B--C--D--cycle); draw(M1--M2--M3--M4--cycle); draw(G1--G2--G3--G4--cycle); dot(P); defaultpen(fontsize(10pt)); draw(A--P--B); draw(C--P--D); label("$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 $\triangle G_1G_2P\sim\triangle M_1M_2P, \triangle G_2G_3P\sim\triangle M_2M_3P, \triangle G_3G_4P\sim\triangle M_3M_4P,$ and $\triangle G_4G_1P\sim\triangle M_4M_1P.$ By the properties of centroids, the side-length ratio for each pair of triangles is $\frac23,$ so the area ratio for each pair of triangles is $\left(\frac23\right)^2=\frac49.$

Let the brackets denote areas. Since $[ABCD]=30^2=900,$ it follows that $\begin{align*} [G_1G_2G_3G_4]&=[G_1G_2P]+[G_2G_3P]+[G_3G_4P]+[G_4G_1P] \\ &=\frac49[M_1M_2P]+\frac49[M_2M_3P]+\frac49[M_3M_4P]+\frac49[M_4M_1P] \\ &=\frac49([M_1M_2P]+[M_2M_3P]+[M_3M_4P]+[M_4M_1P]) \\ &=\frac49[M_1M_2M_3M_4] \\ &=\frac29[ABCD] \\ &=\boxed{\textbf{(C) }200}. \end{align*}$

Remarks for Quadrilateral $\boldsymbol{G_1G_2G_3G_4}$

By angle addition, we have $\begin{align*} \angle G_1G_2G_3 &= \angle G_1G_2P + \angle PG_2G_3 \\ &= \angle M_1M_2P + \angle PM_2M_3 \\ &= \angle M_1M_2M_3 \\ &= 90^\circ. \end{align*}$ Similarly, we deduce that $\angle G_2G_3G_4 = \angle G_3G_4G_1 = \angle G_4G_1G_2 = 90^\circ.$

Note that $M_1M_2M_3M_4$ is a square of side-length $15\sqrt2.$ Since $G_1G_2=\frac23M_1M_2, G_2G_3=\frac23M_2M_3, G_3G_4=\frac23M_3M_4,$ and $G_4G_1=\frac23M_4M_1,$ we deduce that $G_1G_2=G_2G_3=G_3G_4=G_4G_1=10\sqrt2.$

Together, quadrilateral $G_1G_2G_3G_4$ is a square of side-length $10\sqrt2,$ so its area is $\left(10\sqrt2\right)^2=200.$

~RandomPieKevin ~Kyriegon ~MRENTHUSIASM

Solution 2 (Coordinate Geometry)

We put the diagram on a coordinate plane. The coordinates of the square are $(0,0),(30,0),(30,30),(0,30)$ and the coordinates of point P are $(x,y).$ By using the centroid formula, we find that the coordinates of the centroids are $\left(\frac{x}{3},10+\frac{y}{3}\right),\left(10+\frac{x}{3},\frac{y}{3}\right),\left(20+\frac{x}{3},10+\frac{y}{3}\right),$ and $\left(10+\frac{x}{3},20+\frac{y}{3}\right).$ Shifting the coordinates down by $\left(\frac x3,\frac y3\right)$ does not change its area, and we ultimately get that the area is equal to the area covered by $(0,10),(10,0),(20,10),(10,20)$ which has an area of $\boxed{\textbf{(C) }200}.$

Solution 3 (Accurate Diagram)

We can draw an accurate diagram by using centimeters and scaling everything down by a factor of $2.$ The centroid is the intersection of the three medians in a triangle.

After connecting the $4$ centroids, we see that the quadrilateral looks like a square with side length of $7.$ However, we scaled everything down by a factor of $2,$ so the length is $14.$ The area of a square is $s^2,$ so the area is $\boxed{\textbf{(C) }200}.$

Video Solution (Meta-Solving Technique)

https://youtu.be/GmUWIXXf_uk?t=1439

~ pi_is_3.14

@@ Line 85: / Line 85: @@
 &=\boxed{\textbf{(C) }200}.
 \end{align*}</cmath>
+<b>Remarks for Quadrilateral <math>\boldsymbol{G_1G_2G_3G_4}</math></b>
+<ol style="margin-left: 1.5em;">
+  <li>By angle addition, we have
+<cmath>\begin{align*}
+\angle G_1G_2G_3 &= \angle G_1G_2P + \angle PG_2G_3 \\
+&= \angle M_1M_2P + \angle PM_2M_3 \\
+&= \angle M_1M_2M_3 \\
+&= 90^\circ.
+\end{align*}</cmath>
+Similarly, we deduce that <math>\angle G_2G_3G_4 = \angle G_3G_4G_1 = \angle G_4G_1G_2 = 90^\circ.</math>
+</li><p>
+  <li>Note that <math>M_1M_2M_3M_4</math> is a square of side-length <math>15\sqrt2.</math> 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> we deduce that <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=200.</math>
 ~RandomPieKevin ~Kyriegon ~MRENTHUSIASM

2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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