 Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on A Parallelogram and a Line.

## A Parallelogram and a Line – AIME I, 1999

Consider the parallelogram with vertices (10,45),(10,114),(28,153) and (28,84). A line through the origin cuts this figure into two congruent polygons. The slope of the line is $\frac{m}{n}$, where m and n are relatively prime positive integers, find m+n.

• is 107
• is 118
• is 840
• cannot be determined from the given information

### Key Concepts

Parallelogram

Slope of line

Integers

But try the problem first…

Source

AIME I, 1999, Question 2

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

By construction here we see that a line makes the parallelogram into two congruent polygons gives line passes through the centre of the parallelogram

Second Hint

Centre of the parallogram is midpoint of (10,45) and (28,153)=(19,99)

Final Step

Slope of line =$\frac{99}{19}$ then m+n=118.