1951 AMC 12 Problem 48

Question

The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:

1:2

2:3

2:5

3:4

3:5

Let the circle have radius

R

and the semicircle-square have side

s

For the semicircle-square, a top vertex gives

(\frac{s}{2})^2+s^2=R^2

For the square in the semicircle, put its base on the diameter. A top vertex has horizontal distance

\frac{s}{2}

from the center and vertical distance

s,

so

\left(\frac s2\right)^2+s^2=R^2,

giving

s^2=\frac{4R^2}{5}.

A square inscribed in the full circle has diagonal

2R,

hence area

2R^2.

The ratio is

\frac{\frac{4R^2}{5}}{2R^2}=\frac25.

Thus, the correct answer is C.

Problem 48 in other years