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AIME I Algebra Arithmetic Geometry Math Olympiad USA Math Olympiad

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

Check the Answer


Answer: is 315.

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.

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