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


But try the problem first…

Answer: is 820.

Source
Suggested Reading

AIME I, 1992, Question 13

Coordinate Geometry by Loney

Try with Hints


First hint

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

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

Second Hint

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

Final Step

=(10)(82)

=820.

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