The perimeter of a rectangular plot of land is 114 metres and its area is 810 square metres.

(i) Take the length of plot as x metres. Use the perimeter 114 m to write the value of the breadth in terms of x.

(ii) Use the values of length, breadth and area to write an equation in x.

(iii) Solve the equation to find the length and breadth of the plot.

(i) Given,

The length of rectangular plot is x metres

Perimeter of the rectangular plot = 114 m

By formula,

⇒ Perimeter of rectangle = 2(Length + Breadth)

⇒ 114 = 2(x + breadth)

⇒ $\dfrac{114}{2}$ = x + breadth

⇒ Breadth = (57 - x) meters.

Hence, breadth of rectangle = 57 - x.

(ii) Given,

Area of rectangle = 810 m²

By formula,

Area of rectangle = Length × Breadth

⇒ 810 = x(57 - x)

⇒ 810 = 57x - x²

⇒ x² - 57x + 810 = 0

Hence, obtained equation is x² - 57x + 810 = 0.

(iii) Solving,

⇒ x² - 57x + 810 = 0

⇒ x² - 27x - 30x + 810 = 0

⇒ x(x - 27) - 30(x - 27) = 0

⇒ (x - 30)(x - 27) = 0

⇒ (x - 30) = 0 or (x - 27) = 0     [Using zero-product rule]

⇒ x = 30 or x = 27.

Case 1:

If, x = 30 m then breadth = 57 - x = 57 - 30 = 27 m.

Case 2:

If, x = 27 m then breadth = 57 - x = 57 - 27 = 30 m.

Hence, the dimensions of the rectangular plot are 27 m and 30 m.