Question

If the perimeter of a rectangular room is
34 and the length of the diagonal is 13,
then the dimensions of the room are​

Answer 1

Given :

• Perimeter of a rectangular room = 34 units
• Length of diagonal of the room = 13 units

To find :

• The dimensions of the room

Solution :

Using formula,

• Perimeter of rectangle = 2(l + b)

where,

• l denotes the length of the rectangle
• b denotes the breadth of the rectangle

Substituting the given values :

⇒ 34 = 2(l + b)

⇒ 34/2 = l + b

⇒ 17 = l + b ------(1)

Using formula,

• Diagonal of rectangle = √(l² + b²)

Substituting the given values :

⇒ 13 = √(l² + b²)

⇒ Squaring both the sides :

⇒ (13)² = [√(l² + b²)]²

⇒ 169 = l² + b² ------(2)

Taking equation (1) :

⇒ 17 = l + b

⇒ l = 17 - b -----(3)

Substituting equation (3) in (4) :

⇒ 169 = l² + b²

⇒ 169 = (17 - b)² + b²

Using identity :

• (a - b)² = a² + b² - 2ab

⇒ 169 = (17² + b² - 2(17)(b) + b²

⇒ 169 = 289 + b² - 34b + b²

⇒ 169 - 289 = b² + b² - 34b

⇒ - 120 = 2b² - 34b

⇒ 2b² - 34b + 120 = 0

⇒ Taking 2 common :

⇒ 2(b² - 17b + 60) = 0

⇒ b² - 17b + 60 = 0

⇒ A quadratic equation is formed whose product is 60b²

⇒ b² - 12b - 5b + 60 = 0

⇒ b(b - 12) - 5(b - 12) = 0

⇒ (b - 5)(b - 12) = 0

⇒ (b - 5) = 0 or (b - 12) = 0

⇒ b = 5 or b = 12

The dimensions of the rectangular room :

When b = 5 :-

Substituting b = 5 in equation (1) :

→ 17 = l + b

→ 17 = l + 5

→ 17 - 5 = l

→ 12 = l

When b = 12 :-

Substituting b = 12 in equation (1) :

→ 17 = l + b

→ 17 = l + 12

→ 17 - 12 = l

→ 5 = l

Therefore,

• When breadth of the room = 5 units then the length of the room = 12 units
• When breadth of the room = 12 units then the length of the room = 5 units