Question 198156
Remember Rectangles have 4 90 degree angles 2 pair of equal and parallel sides, opposing each other.  Also remember formulas for triangles.  Splitting this rectangle will make things easier.

The width is simply "w".
Length is "7+w=l".
The hypotenuse is 13.

Pythagorean theorem.
{{{a^2+b^2=c^2}}} where a=w b=l and c=13
Since l=7+w we can substitute that value in b.
{{{(w)^2+(7+w)^2=13^2}}}
{{{w^2+49+14w+w^2=169}}}
Put into quadratic form. 
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}  where a=2 b=14 c=-120 x=w
{{{2w^2+14w-120=0}}}
{{{w=(-(14)+-sqrt((14)^2-4(2)(-120)))/(2(2)) }}}
{{{w=(-14+-sqrt(196+960))/4}}}
{{{w=(-14+-34)/4}}}
Split
{{{w=(-14+34)/4}}}  {{{w=(-14-34)/4}}}
{{{w=20/4}}}        {{{w=-48/4}}}
{{{w=5}}}           {{{w=-12}}}

It only makes sense that w=5 is the only answer based on an inability to have a negative measurement.  Substitute w=5 back into l=7+w to get l=12.

Answers
l=12ft w=5ft
P=34ft A=60sqft