1959 AMC 12 Problem 31

Question

A square, with an area of

40,

is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius is:

80

100

120

160

200

Let the first square have side

s

and place its lower side on the semicircle’s diameter

A top vertex has horizontal coordinate

\frac{s}{2}

and vertical coordinate

s

relative to the center

Let the semicircle have radius

r

and the square have side

s,

where

s^2=40.

A top vertex of the square is

\frac{s}{2}

horizontally and

s

vertically from the center, so

r^2=s^2+\left(\frac s2\right)^2=\frac54s^2=50.

A square inscribed in the full circle has diagonal

2r,

hence area

2r^2=100.

Thus, the correct answer is B.

Problem 31 in other years