Division of Square Inspired

Geometry Level 3

A straight line divides a unit square into two regions such that the bigger region is double the area of the smaller region. This line also intercepts the square at the midpoint of one of the sides. It intercepts the square again at the opposite side of the square and divides this side into two parts. Given that the longer part of this side is a b \frac{a}{b} where a a and b b are positive coprime integers, a + b = ? a+b=\boxed{?}


The answer is 11.

Noel Lo by Noel Lo , 24, Singapore

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Noel Lo
Jul 26, 2017

The straight line divides the square into two trapeziums. It cuts one of the sides into two parts each of length 1 2 \frac{1}{2} and the opposite side into two parts of lengths x x and ( 1 − x ) (1-x) where x x is the longer part as required. Comparing the areas of the two regions:

x + 1 2 2 = 2 ( 1 − x + 1 2 2 ) \frac{x+\frac{1}{2}}{2}=2(\frac{1-x+\frac{1}{2}}{2})

x + 1 2 = 2 ( 1 + 1 2 − x ) x+\frac{1}{2}=2(1+\frac{1}{2}-x)

x + 1 2 = 2 + 1 − 2 x x+\frac{1}{2}=2+1-2x

x + 1 2 = 3 − 2 x x+\frac{1}{2}=3-2x

x + 2 x = 3 − 1 2 x+2x=3-\frac{1}{2}

( 1 + 2 ) x = 6 − 1 2 (1+2)x=\frac{6-1}{2}

3 x = 5 2 3x=\frac{5}{2}

x = 5 3 × 2 x=\frac{5}{3\times2}

x = 5 6 x=\frac{5}{6}

Therefore, a + b = 5 + 6 = 11 a+b=5+6=\boxed{11}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...