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Largest Area of Triangle | AIME I, 1992 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Largest Area of Triangle.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Largest Area of Triangle.

Area of Triangle – AIME I, 1992


Triangle ABC has AB=9 and BC:AC=40:41, find the largest area that this triangle can have.

  • is 107
  • is 820
  • is 840
  • cannot be determined from the given information

Key Concepts


Ratio

Area

Triangle

Check the Answer


Answer: is 820.

AIME I, 1992, Question 13

Coordinate Geometry by Loney

Try with Hints


Let the three sides be 9, 40x, 41x

area = \(\frac{1}{4}\sqrt{(81^2-81x^2)(81x^2-1)} \leq \frac{1}{4}\frac{81^2-1}{2}\)

or, \(\frac{1}{4}\frac{81^2-1}{2}=\frac{1}{8}(81-1)(81+1)\)

=(10)(82)

=820.

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