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Area Ascertainment II (Posted on 2014-10-24) Difficulty: 3 of 5
In triangle ABC, it is known that AB = 9 and BC/AC = 40/41.

Find the maximum possible area of the triangle ABC.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 2 of 8 |

Let A = (-c,0), B = (c,0), and C = (x,y)

   where c = 9/2.

    |BC|     sqrt[(x - c)^2 + y^2]
   ------ = ----------------------- = k = 40/41.
    |AC|     sqrt[(x + c)^2 + y^2]

          or

   (x - c)^2 + y^2 = k^2*[(x + c)^2 + y^2]     (1)

taking the derivative of (1) with respect
to x and setting dy/dx = 0 gets

   x = c*(1 + k^2)/(1 - k^2).

Plugging x into (1) and solving for y:

   y = 2*c*k/(1 - k^2)

   max. area of triangle ABC = c*y 

                             = 2*c^2*k/(1 - k^2) 

                             = 820

                      

  Posted by Bractals on 2014-10-24 18:28:09
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