If the perimeter of a rectangular plot is 68 m and length of its diagonal is 26 m, find its area.

Let ABCD be a rectangular plot of length x m and breadth y m.

By formula,

Perimeter = 2(length + breadth)

Substituting the values we get,

⇒ 68 = 2(x + y)

⇒ 34 = x + y

⇒ x = 34 - y ……… (1)

In right angle triangle ABC

⇒ AC² = AB² + BC² (By pythagoras theorem)

⇒ 26² = x² + y²

⇒ x² + y² = 676

Substituting the value of x from equation (1),

⇒ (34 – y)² + y² = 676

⇒ 1156 + y² – 68y + y² = 676

⇒ 2y² – 68y + 1156 – 676 = 0

⇒ 2y² – 68y + 480 = 0

⇒ 2(y² – 34y + 240) = 0

⇒ y² – 34y + 240 = 0

⇒ y² – 24y – 10y + 240 = 0

⇒ y(y – 24) – 10(y – 24) = 0

⇒ (y – 10)(y – 24) = 0

⇒ y – 10 = 0 or y – 24 = 0

⇒ y = 10 m or y = 24 m.

Now substituting the value of y in equation (1)

⇒ y = 10 m, x = 34 – 10 = 24 m

⇒ y = 24 m, x = 34 – 24 = 10 m

Area in both cases = xy

= 24 × 10 or 10 × 24

= 240 m².

Hence, the area of the rectangular block is 240 m².