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?

$100\sqrt{2}$

$100\sqrt{3}$

$200$

$200\sqrt{2}$

$200\sqrt{3}$

Answer: C

Concepts:centroid coordinate geometry area

Difficulty rating: 1810

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.

2018 AMC 12B Problem 13

Answer: C

Difficulty rating: 1810

Solution:

Problem 13 in Other Years