Question

The diagonals of a rhombus are in the ratio of 3:4, and the area is 54cm². Find the side of the rhombus​

Answer 1

Side of rhombus is 7.5 cm

Given :-

The diagonals of a rhombus are in the ratio of 3:4, and the area is 54cm²

To Find :-

Side

Solution :-

Let the rhombus be ABCD

AC & BD are diagonals

Now,

Let

AC = 3x & BD = 4x

Now,

We know that

${\large{\boxed{\underline{\bf Area_{(Rhombus)}=\dfrac{1}{2}\times D_1\times D_2}}}}$

=> 54 = ½ × 3x × 4x

=> 54 = ½ × 12x²

=> 54 = 6x²

=> 54/6 = x²

=> 9 = x²

=> √(9) = x

=> 3 = x

Now,

AC = 3(3) = 9 cm

BD = 4(3) = 12 cm

Let the intersecting point of the diagonals be O

We know that diagonal of rhombus bisect each other at right angles. So,

In ∆AOB

=> AO = AC/2 = 9/2 = 4.5 cm

=> OB = BD/2 = 12/2 = 6 cm

Now,

=> (AB)² = (AO)² + (OB)² [Pythagoras theorem]

=> (AB)² = (4.5)² + (6)²

=> (AB)² = 20.25 + 36

=> (AB)² = 56.25

=> AB = √(56.25)

=> AB = 7.5 cm

Therefore,

Side of square is 7.5