1989 AMC 12 Problem 21

Question

A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up

36\%

of the area of the flag, what percent of the area of the flag is blue?

0.5

1

2

3

6

Answer: C

Concepts:area decomposition percentage symmetry

Difficulty rating: 1980

Small hint:

Let

h

be half a diagonal of the central blue square after scaling the flag side to

1

Big hint:

The four white corner triangles together have area

(1-2h)^2

Solution:

Scale the flag to a unit square, and let

h

be the horizontal distance from its center to a vertex of the blue square. The four congruent white corner triangles have total area

(1-2h)^2.

Since the cross occupies

0.36,

(1-2h)^2=0.64.

With

0\lt h\lt\frac{1}{2},

this gives

h=0.1.

The blue square has perpendicular diagonals

2h

and

2h,

so its area is

\frac12(2h)^2=2h^2=0.02,

or

2\%

of the flag.

Thus the correct answer is C.