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.

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.

Ratio

Area

Triangle

Answer: is 820.

AIME I, 1992, Question 13

Coordinate Geometry by Loney

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.