Question

One of the diagonals of a rhombus is thrice as the other. The
area of the rhombus is 54 sq.cm. the sum of the length of the
diagonals.

Answer 1

Answer :

The sum of the length of the diagonals is 24 cm.

Given :

• One of the diagonals of a rhombus is thrice as the other.

• Area of the rhombus is 54 cm².

To calculate :

• The sum of the length of the diagonals.

Calculation :

Let us assume the first diagonal of the rhombus as "x". Clearly, the second diagonal becomes "3x".

We have,

• First diagonal (D₁) ⇒ x cm
• Second diagonal (D₂) ⇒ 3x cm

Calculating the length of both diagonals:

We know that,

$\longrightarrow$ Area of rhombus : ½ × diagonal₁ × diagonal₂

According to the question,

$\longrightarrow$ 54 = ½ × x × 3x

$\longrightarrow$ 54 = ½ × 3x²

$\longrightarrow$ 54 × 2 = 1 × 3x²

$\longrightarrow$ 54 × 2 = 3x²

$\longrightarrow$ 108 = 3x²

$\longrightarrow$ $\sf \dfrac{108 }{3}$ = x²

$\longrightarrow$ 36 = x²

$\longrightarrow$ √36 = x

$\longrightarrow$ 6 = x

Henceforth,

$\longrightarrow$ First diagonal = x cm

$\longrightarrow$ First diagonal = 6 cm

$\longrightarrow$ Second diagonal = 3x cm

$\longrightarrow$ Second diagonal = 3(6) cm

$\longrightarrow$ Second diagonal = 18 cm

Calculating sum of the diagonals :

$\longrightarrow$ Sum = First diagonal + Second diagonal

$\longrightarrow$ Sum = 6 cm + 18 cm

$\longrightarrow$ Sum = 24 cm

Therefore, sum of the length of the diagonals is 24 cm.

More about rhombus :

• Perimeter of rhombus = 4 × Side

• Area of rhombus = ½ × diagonal₁ × diagonal₂

• Diagonals of the rhombus bisect each other.

• Diagonals bisect each other at right angles.

• A rhombus is a parallelogram.

• Opposite angles of a rhombus are equal.

• All the sides of the rhombus are equal.

Answer 2

Let the length of diagonal 1 be = (d)

∴ Length of diagonal 2 = (d/3)

Area = 54 cm^2

We know, area of a rhombus = ½ · d1 · d2

According to the Question,

½ × d × d/3 = 54

⇒ d^2/6 = 54

⇒ d^2 = 54 × 6

⇒ d = √(54 × 6) = √(3^2 × 2 × 3 × 2 × 3)

⇒ d = √(3^2 × 2^2 × 3^2)

⇒ d = 3 × 2 × 3 = 18 cm.

So, d1 + d2 = 18 cm + ⅓(18 cm) = 18 cm + 6 cm = 24 cm.

∴ The sum of the diagonal is 24 cm.