2018 AMC 12B Problem 13

Problem 13 of 25IntermediateGeometry

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?

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Solution

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.