Find Green and Magenta area ratio in Given figure | Figure contain 4 squares

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In this figure One square is drawn. From it's one vertices another square is drawn with some inclination. From vertices of both squares Two squares are drawn. One is bigger and one is smaller. Find ratio of areas of both Squares.

Step 1:

Let green square is of side length x and magenta square is with side length y.

Green area is x²

Magenta area is y²

We have to find x²/y²

Step 2:

Let side length of two small squares are a and b.

Inclination angle between two squares is θ.

Step 3:

Consider triangle AOC.

Angle O = 90+θ

Apply cosine rule on angle O.

y²=a²+b²-2×a×b×cos(90+θ) --------(1)

Step 4:

Consider triangle POQ.

Angle O = 90+θ

Apply cosine rule on angle O.

x²=2a²+2b²-4×a×b×cos(90+θ)

x²=2(a²+b²-2×a×b×cos(90+θ)) --------(2)

Step 5:

From (1) and (2)

x²/y² = 2(a²+b²-2×a×b×cos(90+θ)) / a²+b²-2×a×b×cos(90+θ)

x²/y² = 2

In this way Green:Magenta ratio is 2:1