[Math] How to get the shaded region of the rectangle

Question

I have this problem:

So my development was:

Denote side of rectangle with: $2a, 2b$.

So, $4ab= 64, ab = 16$

Denote shaded region with $S$

Denote area of triangle $DGH = A_1$ and triangle $FBE = A_2$.

So, $A_1 + A_2 + S = 64$

$S = 64 – A_1 – A_2$

The triangles $A_1, A_2$ are congruent because $LAL$ congruence criterion.

The area of $A_1$ and $A_2$, is the same and i got it with this way:

Since, the $\angle{GDH} = 90$ and the median from this angle to the base $HG$, that is the altitude of the triangle $DGH$, will measure the half of the $HG$ side.

And the $HG$ side by Pythagorean theorem, will be $\sqrt{a^2 + b^2}$, that will be the base of the triangle.

And the altitude will be: $\frac{\sqrt{a^2 + b^2}}{2} $,

So the Area of $A_1 = \frac{a^2 + b^2}{4}$

So, $A_1 + A_2 = \frac{a^2 + b^2}{2}$

Then, $64 – (\frac{a^2 + b^2}{2}) = S$

And, $-(a^2 – 8ab + b^2) = 2S$

And I have not been able to continue from here, what should I do? Thanks in advance.

Answer 1

Hint:

• there are eight small red triangles all with the same area

• six of them are shaded