Answer
The dimension of the rectangle is $4\; feet $ by $3\; feet $.
Work Step by Step
Let the length of the rectangle is $l$.
and width of the rectangle is $w$
Diagonal is $D=5\; feet$.
and perimeter is $P=14\; feet$.
Formula for the diagonal is
$D^2=l^2+w^2$
Substitute values.
$5^2=l^2+w^2$
$25=l^2+w^2$ ... (1)
The formula for the perimeter is
$P=2(l+w)$
Substitute values.
$14=2(l+w)$
$7=l+w$ ... (2)
Square both side.
$49=l^2+w^2+2lw$
From equation (1).
$49=25+2lw$
$49-25=2lw$
$24=2lw$
$12=lw$
$\frac{12}{w}=l$
Substitute into equation (2).
$7=\frac{12}{w}+w$
Multiply by $w$.
$7w=12+w^2$
$w^2-7w+12=0$
Factor the equation.
$w^2-4w-3w+12=0$
$w(w-4)-3(w-4)=0$
$(w-4)(w-3)=0$
The possible values of $w$ is
$w=3,4$.
Substitute back into equation (2).
$7=l+3$
$7-3=l$
$4=l$
and
$7=l+4$
$7-4=l$
$3=l$.
