# Adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of the sides of this parallelogram. Show that its diagonals are in the ratio square root of 3 :1.

A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal.

## Answer: AC : BD = ⎷3 : 1

A parallelogram whose adjacent sides are equal is known as a rhombus.

**Explanation:**

Let's draw the diagram of the parallelogram ABCD along with its diagonals as shown below:

As, ABCD is a parallelogram,

Therefore, AB=CD and BC=AD

Now, according to the question AB=BC,

Therefore, AB=BC=CD=AD

Hence, ABCD is a rhombus

Now, as one of the diagonals is equal to its sides, therefore, AB=BC=CD=AD=BD

Let's assume AB=BC=CD=AD=BD=a

As, BD=a,

=> BO=a/2 (Since, diagonals bisect each other at right angle in a rhombus)

Hence, △AOB is right-angled at O.

Now, apply the Pythagoras Theorem on △AOB

=> AB^{2} = BO^{2} + AO^{2}

=> a^{2} = (a/2)^{2} + AO^{2}

=> AO^{2} = a^{2} - a^{2}/4

=> AO^{2 }= 3a^{2}/4

=> AO = a⎷3/2

As AO = a⎷3/2 therefore AC = 2AO = a⎷3

Therefore, the length of the diagonals are AC = a⎷3 and BD = a

The ratio of the diagonals are:

AC/BD = a⎷3/a

= ⎷3/1