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April 22, 2020

A Parallelogram and a Line | AIME I, 1999 | Question 2

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


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.

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