enter image description here

In the right angled triangle $ABC$, $\angle A = 90^\circ$, $AB=8$, $AC=6$, $BC = 10$. $D$ is a point on $AB$ in such way that if a perpendicular $DE$ is drawn on $BC$ from $D$ then $BE = 4$. What is the area of the quadrilateral $ADEC$?

Source: Bangladesh Math Olympiad 2015 junior category.

I can not find any way to solve this math. Can anyone give me a hint?

  • 4
    $\begingroup$ Hint: as  = Ê = 90 and B = B, ABC and EBD are similar. That should do the trick. $\endgroup$ – Jonas De Schouwer Feb 14 at 17:40
  • $\begingroup$ I was just about to write the same. $\endgroup$ – Michael Hoppe Feb 14 at 17:46
  • 1
    $\begingroup$ Further hint (if necessary): Since the triangles are similar, and you know all of the side of the larger, and one of the sides of the smaller, you can deduce all of the sides of the smaller. Thus you can compute the areas of the larger and smaller triangles. Their difference is the desired area. $\endgroup$ – Keith Backman Feb 14 at 18:38

It is just the sum of the area of two equal right triangles: enter image description here


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