Question 821306
{{{drawing(300,240,-1.5,13.5,-1.5,10.5,
rectangle(0,0,12,9),locate(6,4,15),
rectangle(0.5,0.5,0,0),green(line(0,9,12,0)),
locate(5.5,0,x),locate(0.2,5,9)
)}}} The diagonal splits the rectangle into two congruent right triangles.
In each of those triangles, the hypotenuse is the diagonal of the rectangle,
one leg is the width of the rectangle,
and the other leg in the length of the rectangle.
If the length is {{{x}}}{{{feet}}} , according to the Pythagorean theorem,
{{{x^2+9^2=15^2}}}
{{{x^2+81=225}}}
{{{x^2=225-81}}}
{{{x^2=144}}}
{{{x^2=12^2}}}
So the length of the rectangle is {{{12 feet}}} .
{{{x=-12}}} is another solution of {{{x^2=12^2}}} , but lengths are never negative numbers.
 
The area of a rectangle with a length of {{{12 feet}}} and a width of {{{9 feet}}} is
{{{(9feet)(12 feet)=highlight(108)}}}{{{square}}}{{{feet}}} .