Let the square side be r.
Therefore r is the length of the red rectangle and also the radius of the inscribed quarter circle.
Let the base of the red area be a.
b = √(16-a²)
Therefore;
r² = (r-a)²+b²
r² = r²-2ar+a²+16-a²
2ar = 16
ar = 8
Therefore red area/area red rectangle is ar.
= 8 square units.
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