The area of the semicircle in Figure A is half the area of the circle in Figure B. The area of a square inscribed in the semicircle, as shown, is what fraction of the area of a square inscribed in the circle? Express your answer as a common fraction.
+128474 Call the radius of the semi-circle and the circle, r
Call the side of the small square, s
So...using the Pythagorean Theorem, we have that
(1/2)s^2 + s^2 = r^2
(1/2)s^2 + s^2 = r^2
(5/4)s^2 = r^2
√5/2 s = r
s = 2r / √5
So...the area of the small square = s^2 = (4/5)r^2
And..in the larger circle....the side of the square is √2r
So....the area of the square in the larger circle is just 2r^2
So....the area of the smaller square's area to the larger square's area is
(4/5)r^2 / 2 r^2 =
4/10 =
2/5 of the large square's area
