2007 AMC 10A Problem 19

Problem 19 of 25HarderAlgebra Geometry

A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

Show solution

Solution

Let

s

be the side,

w

the brush width, and

x

the leg of one unpainted isosceles right triangle. Each triangle has area

\tfrac18 s^2,

\tfrac12 x^2 = \tfrac18 s^2

and

x = \tfrac{s}{2}.

The leg plus the brush width is half the diagonal:

x + w = \tfrac{\sqrt2}{2} s.

Thus

w = \tfrac{\sqrt2}{2} s - \tfrac{s}{2}.

Therefore

\dfrac{s}{w} = \dfrac{2}{\sqrt2 - 1} = 2\sqrt2 + 2.

Thus, the correct answer is C.

Answer choices

Solution

More practice