## Four Squares

A rectangle with side lengths $a$ and $20$ is divided into $4$ squares as shown. What is the value of $a$?

Source: NCTM Mathematics Teacher, December 2005

SOLUTION

Sum of the areas of the $4$ squares equals the area of the rectangle
$a^2+(2x)^2+x^2+x^2=20a$
$a^2+6x^2=20a$
$(3x)^2+6x^2=20(3x)$
$9x^2+6x^2=60x$
$15x^2-60x=0$
$15x(x-4)=0$
$x=0$ (extraneous solution) or $x=4$
$a=3x=3(4)=12$

Answer: $12$ units

Alternative solution
Sum of a side length of the large square and a side length of the medium square equals the length of the rectangle
$3x+2x=20$
$5x=20$
$x=4$