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

Check the Answer


But try the problem first…

Answer: is 315.

Source
Suggested Reading

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.

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