Question
Mon October 22, 2012 By:

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

Expert Reply
Tue October 23, 2012
In a semicircle of radius r, an inscribed square will touch the circumference at two points symmetrically placed about the centre of the base. Drawing the radius to one of these points will produce a triangle with base s/2 and height s, where s is the length of the sides of the square. By Pythagoras we have r² = s² + (s/2)² = (5/4)s² and hence the area of the square s² = (4/5)r²

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.
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