I am struggling on this question where I don't understand how to work out the area of the red square in the question. I have shown my workings in my question and is it possible if I could be assisted with this question and tell me where I went wrong.
Thank You
On
You have two problems. The side of the red square is not $10$. The top side of the square is divided evenly, so each half is $10$. The side of the red square is longer than $10$. What is the length of the diagonal lines? Pythagoras can help there. Now draw a line from the midpoint of the top parallel to the diagonal line that is the top side of the red square. You get a similar triangle.
After that, you had no reason to subtract $100$ from $400$. If the side of the red square had been $10$ the answer would be $100$.
On
The large square is made up of five of the red square, since each quadrilateral and small right triangle make one red square and there are four of these pairs within the large square. So the area of the red square is one-fifth of $400$, or $80$. You can check this. Since $20^2+10^2=500$, the long line within the big square is $$\sqrt 500\approx 22.36=2\times \sqrt 80+\sqrt 20\approx 2\times 8.94+4.47$$
As you can see, the large square can be decomposed into $20$ equal triangles, each one of area $20$. The central square contains four of them, hence its area is $80$.