2007 AMC 12A Problem 19

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

Answer: E

Concepts:coordinate geometry triangle area

Difficulty rating: 1840

Small hint:

The area of

\triangle ABC

fixes the altitude from

A

to

BC,

so

A

lies on

y=18

or

y=-18

Big hint:

Likewise

A

lies on one of two lines parallel to

DE

; the four intersection points average to the center of a parallelogram

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.