2018 AMC 12B Problems/Problem 13

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 (Drawing an 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: $\boxed{\textbf{(C) } 200}.$

Solution 2

The centroid of a triangle is $\frac{2}{3}$ of the way from a vertex to the midpoint of the opposing side. Thus, the length of any diagonal of this quadrilateral is $20$ . The diagonals are also parallel to sides of the square, so they are perpendicular to each other, and so the area of the quadrilateral is $\frac{20\cdot20}{2} = 200$ , $\boxed{(C)}$ .

Solution 3

The midpoints of the sides of the square form another square, with side length $15\sqrt{2}$ and area $450$ . Dilating the corners of this square through point $P$ by a factor of $3:2$ results in the desired quadrilateral (also a square). The area of this new square is $\frac{2^2}{3^2}$ of the area of the original dilated square. Thus, the answer is $\frac{4}{9} * 450 = \boxed {C}$

Solution 4

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 $(\frac{x}{3},10+\frac{y}{3}),(10+\frac{x}{3},\frac{y}{3}),(20+\frac{x}{3},10+\frac{y}{3}),$ and $(10+\frac{x}{3},20+\frac{y}{3}).$ Shifting the coordinates down by $(\frac x3,\frac y3)$ 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 $\boxed{(C) 200.}$

Solution 5 Lazy solution

Since the problem does not constrain $P$ , it indicates that the result does not depend on the location of $P$ . So let's put it in the center. Then it becomes obvious the quadrilateral is a square side length equal 2/3 of the diagonal of each quarter of the original square, which is $\frac{2}{3}*15\sqrt{2}=10\sqrt{2}$ , so area is $\boxed{(C) 200.}$

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

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