Categories

# Points of Equilateral triangle | AIME I, 1994 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.

## Points of Equilateral triangles – AIME I, 1994

The points (0,0), (a,11), and (b,37) are the vertices of equilateral triangle, find the value of ab.

• is 107
• is 315
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

Complex Number

Equilateral Triangle

AIME I, 1994, Question 8

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

Let points be on complex plane as b+37i, a+11i and origin.

then $(a+11i)cis60=(a+11i)(\frac{1}{2}+\frac{\sqrt{3}i}{2})$=b+37i

equating real parts b=$\frac{a}{2}-\frac{11\sqrt{3}}{2}$ is first equation

equating imaginary parts 37=$\frac{11}{2}+\frac{a\sqrt{3}i}{2}$ is second equation

solving both equations a=$21\sqrt{3}$, b=$5\sqrt{3}$

ab=315.