Question

Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0),$ $C=(223,0),$ $D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$ -coordinates of $A?$

$282$

$300$

$600$

$900$

$1200$

Solution:

The altitude $h$ from $A$ in $\triangle ABC$ satisfies $2007=\tfrac12\cdot 223\cdot h,$ so $h=18.$ Thus $A$ lies on $y=18$ or $y=-18.$

Line $DE$ has equation $x-y-300=0.$ The condition on $\triangle ADE$ similarly places $A$ on one of two lines parallel to $DE.$

The four possible positions of $A$ are the vertices of a parallelogram whose center is the intersection of $y=0$ with line $DE,$ namely $(300,0).$ Hence the sum of the four $x$ -coordinates is $4\cdot 300=1200.$

Thus, the correct answer is E.

2007 AMC 12A Problem 19

Solution:

Problem 19 in Other Years