Question

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?

$2\sqrt{2} + 1$

$3\sqrt{2}$

$2\sqrt{2} + 2$

$3\sqrt{2} + 1$

$3\sqrt{2} + 2$

Solution:

Let $s$ be the side, $w$ the brush width, and $x$ the leg of one unpainted isosceles right triangle. Each triangle has area $\tfrac18 s^2,$ so $\tfrac12 x^2 = \tfrac18 s^2$ 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.

2007 AMC 10A Problem 19

Solution:

Problem 19 in Other Years