2018 AMC 12B Problem 13

Question

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?

100\sqrt{2}

100\sqrt{3}

200

200\sqrt{2}

200\sqrt{3}

Place the square in coordinates and write

P=(3x,3y)

Each centroid averages three vertices; the four centroids form a square whose diagonals have length

20

Place

A=(0,30),

B=(0,0),

C=(30,0),

D=(30,30),

and

P=(3x,3y).

Averaging the vertices, the four centroids are

\begin{gathered} (x,\,y+10),\ \\ (x+10,\,y),\ \\ (x+20,\,y+10),\ \\ (x+10,\,y+20). \end{gathered}

These form a square whose diagonals, one horizontal and one vertical, each have length

20.

Its area is

\tfrac12\cdot20\cdot20=200,

independent of where

P

lies.

Thus, the correct answer is C.

Problem 13 in other years