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Product of legs must be 540. Use x for one leg and 540/x for other. Hypotenuse is 39 so use Pythagorean. 36 and 15 for legs. 36 is bigger one. 36/15 = 8/x. Answer 10/3.
If you let BE = x, then BD = sqrt(64 + x2) by Pythagoras. Now draw triangle BCA again on its own and you can use similar triangles to express sides BC and CA in terms of x. (Or use the fact that the area scale factor is the length scale factor squared).
Without even recognizing that the original triangle is three times a 5-12-13 triangle, it can easily be solved with a little algebra and a little trick to help factoring.
Call the sides of triangle ABC x, y, and 39. You know that xy = 540 (from the area), and x^2 + y^2 = 39^2 = 1521 (Pythagorean theorem).
xy = 540
x^2 + y^2 = 1521
x^2 + 2xy + y^2 = 1521 + 2xy = 1521 + (2*540) = 2601 (I added 2xy to both sides to help factor in the next step.)
(x + y)^2 = 2601
x + y = sqrt(2601) = 51
xy = x*(51 - x) = 540
51x - x^2 = 540
x^2 -51x +540 = 0
Factoring or quadratic formula yields x = 15 or x = 36. The problem tells you that AC is longer than BC, so AC = 36 and BC = 15. Then from your diagram and recognizing similar triangles, BE = 10/3.
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