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: If adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of the sides of the parallelogram, then 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, in a rhombus diagonals bisect each other at right angle)

Hence, △AOB is right-angled at O.

Now, apply the Pythagoras theorem on △AOB,

⇒ AB² = BO² + AO²

⇒ a² = (a/2)² + AO²

⇒ AO² = a² - a²/4

⇒ AO² = 3a²/4

⇒ AO = a⎷3/2

As AO = a⎷3/2, therefore AC = 2AO = a⎷3 units.

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

The ratio of the diagonals is:

AC/BD = a⎷3/a

= ⎷3/1

Therefore, AC : BD = ⎷3 : 1.