Area

Let the width of the house is $4x$ and the length is $3x$ .

Therefore,

$\begin{aligned}2(4x+3x)&=140\\14x&=140\\x&=10\end{aligned}$

Hence, the area of the house is,

length $\times$ width

$\begin{aligned}&=(4x)\times(3x)\\&=3\times4\times x^2\\&=12x^2\\&=12\times10^2\\&=12\times 100\\&=\boxed{1200}\end{aligned}$

The ratio of the width to length is $4:3$ . So, we have

$\dfrac{W}{L}=\dfrac{4}{3}$ or $W=\dfrac{4}{3}L$

The perimeter is the sum of the lengths of the sides. So, we have

$P=2L+2W$

$140=2L+2(\dfrac{4}{3}L)$

$140=2L+\dfrac{8}{3}L$

$140=\dfrac{6}{3}L+\dfrac{8}{3}L$

$140=\dfrac{14}{3}L$

$(140)(3)=14L$

$420=14L$

Dividing both sides by $14$ , we get

$30=L$

It follows that

$W=\dfrac{4}{3}(30)=40$

Finally,

$A=L\times W=30\times40=$ $\boxed{1200~cm^2}$ $\color{#20A900}\large\boxed{\text{answer}}$