geometry puzzle (21)


This puzzle was found in FaceBook group "geometria":
(by Eldeniz Hesenov)



We see a rectangle and within two squares with areas 25 and 24.

Asked: calculate the size of the yellow colored area.

The approach

Best is to calculate the rectangle area then subtract 25 and 24.
Base and height of the reactangle are unknown.
Calling these x and y will need two equations to solve.
However, we notice all similar right angled triangles.
This angles define all other dimensions.
So, as a single variable we choose an angle, see picture.



Call this angle α
Given AF=5, AD may be calculated.
Similar BC = BG + GC may be calculated.
At this moment we have two formulas with α which are equal because AD = BC.

Solution




In this equation α has to be solved.
This caused me some troubles, so first I looked for a graphical solution.
In Graphics-Explorer two equations were added:
(x is angle in degrees, y is length)
    formula 1: y = 1 + tan(x)
    formula 2: y = sqrt(24/25)*(sin(x) + 1/cos(x))
At the right top of the Graphics-Explorer screen: selection of degrees, not radians.
Plot the formulas, zoom in to the intersection and read that x = 15.
Unknow at this point is if this answer is correct or just an approximation.

The equation

First we simplify the formula using only the sin and cos functions.
Then the cos(α) denominators are removed.



Now we apply a trick substitution: sin(α)+cos(α)=y

The result is a second degree equation of y.
Below is the proof that this substitution is correct:



Continuing....
For clarity we temporary write p for the square root of 24/25.
Now apply the ABC rule to solve 2nd degree equations:



Some standard trigonometric rules are needed:
Please note: cos(30) = y/(2sin(45))



The same answer as we obtained graphical.

The last part:

AD = AF + FD = 5 + 5.tan(15) = 6,34....
AB = 5 + 5/tan(15) = 23,66.....
AB*AD = 150.
The yellow colored area is 150-25-24 = 101.

A very nice problem.