following are the given options
what is the ratio of the area of a square inscribed in a semi circle to the area inscribed in a quadrant of the same circle
In a quadrant of a circle with centre O, the inscribed square has one corner at O and sides of equal lengths s such that the diagonally opposite corner touches the circumference at point A. Applying Pythagoras to the radius OA gives s² + s² = r², so that the area of this square is s² = r²/2.
Hence the ratio of the area of the square inscribed in a semicircle to that inscribed in a quadrant of the same radius is (4/5)/(1/2) = 8/5 that is, last option.