Difference between revisions of "2018 AMC 12B Problems/Problem 13"
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| − | + | ==Problem== | |
| + | |||
| + | Square <math>ABCD</math> has side length <math>30</math>. Point <math>P</math> lies inside the square so that <math>AP = 12</math> and <math>BP = 26</math>. The centroids of <math>\triangle{ABP}</math>, <math>\triangle{BCP}</math>, <math>\triangle{CDP}</math>, and <math>\triangle{DAP}</math> 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> | ||
| + | |||
| + | |||
| + | <math>\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}</math>[/quote] | ||
| + | |||
| + | ==Solution== | ||
Revision as of 17:50, 16 February 2018
Problem
Square 
has side length 
. Point 
lies inside the square so that 
and 
. The centroids of 
, 
, 
, and 
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]](http://latex.artofproblemsolving.com/4/a/7/4a7552d7af26cb2be88658376dd5934a70ef16c6.png)
[/quote]
