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# 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.

## 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

First hint

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

Second Hint

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

Final Step

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.