Question

The length of a rectangle is thrice as long as the side of a square. The side of the square is 4 cm more than width of a the rectangle. Their areas being equal, find their dimensions.

Answer 1

$$\begin{gathered} \text{Let the breadth of rectangle be}x\text{cm}\text{.} \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}: \\ \text{Side of the square = (}x+4)\text{cm} \\ \text{Length of the rectangle = {3}(x+4)\text{} cm} \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\text{the areas of the rectangle and square are same.} \\ \therefore 3(x+4)\times x={(x+4)}^2 \\ \Rightarrow 3x^2+12x={(x+4)}^2 \\ \Rightarrow 3x^2+12x=x^2+8x+16 \\ \Rightarrow 2x^2+4x-16=0 \\ \Rightarrow x^2+2x-8=0 \\ \Rightarrow x^2+(4-2)x-8=0 \\ \Rightarrow x^2+4x-2x-8=0 \\ \Rightarrow x(x+4)-2(x+4)=0 \\ \Rightarrow (x+4)(x-2)=0 \\ \Rightarrow x=-4\text{or}x=2 \\ \therefore x=2(\because \text{The value of}x\text{cannot be negative)} \\ \mathrm{Thus},\mathrm{t}\text{he breadth of the rectangle is 2 cm and length is {3}(2+4)\text{=18} cm.} \\ \mathrm{Also},\text{the side of the square is 6 cm.} \end{gathered}$$