Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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.

Subscribe to Cheenta at Youtube


Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com