Geometry without using trigonometry
So I'm supposed to find the area of the blue figure without using trigonometry. What do I have to do to find this?
Answer
Look at the attached picture. We have that the triangles $ABC$ and $ADE$ are similar, so since $AE = 2AD$ then $AC = 2AB$. Now since there are so many right angles, all the segments marked with a single tick have the same length, and the ones marked with two ticks have twice that length. It follows that, if the big square has side lenght equal to $1$, then $AB = 1/6$. Now we can split the blue figure in four triangles, whose areas are easy to calculate: the green ones have area $1/9$, the pink ones $1/18$. Putting all those together, the area of the blue figure is \[ 2 \cdot \frac{1}{ 9} + 2 \cdot \frac{1}{18} = \frac{3}{9} = \frac{1}{3}. \]

I obviously meant BC instead of AB and DE instead of AD, but I guess you figured it out!
 answered
 1749 views
 $5.89
Related Questions
 Find the area bounded by the graphs of two functions

Find a general solution for the lengths of the sides of the rectangular parallelepiped with the
largest volume that can be inscribed in the following ellipsoid  Can enough pizza dough be made to cover the surface of the earth?
 How do I negate the effect that the rotation of a plane has on a rectangle?
 Geometry Problem about Hole Placement on PVC Pipe
 Determine formula to calculate the radii of a unique ellipsoid from coordinates of noncoplanar locii on its surface, and without knowing its center or rotation angles.
 Determine the surface area of a ball, rotating a function about $x$axis.
 Direction of forces acting on points A, B and C
Low bounty! Your deadline is also too short.
I think there's some missing info? Do the diagonal lines pass through the middle points of the edges of the square?