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

## Check the Answer

But try the problem first…

Answer: is 118.

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.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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