Adjacent Sides Of A Parallelogram

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 :one.

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

Respond: If adjacent sides of a parallelogram are equal and one of the diagonals is equal to any ane of the sides of the parallelogram, and then Ac : BD = ⎷3 : i.

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

Caption:

Permit'southward draw the diagram of the parallelogram ABCD along with its diagonals every bit shown below:

Parallelogram ABCD with side AB=a

As ABCD is a parallelogram,

Therefore, AB=CD and BC=Advertising

At present, 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=Advertising=BD

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

Every bit, BD=a,

⇒ BO=a/ii (Since, in a rhombus diagonals bisect each other at correct angle)

Hence, △AOB is correct-angled at O.

Now, employ the Pythagoras theorem on △AOB,

⇒ AB² = BOⁱⁱ + AO²

⇒ a^two = (a/2)² + AO²

⇒ AO² = aⁱⁱ - a²/4

⇒ AO²= 3a²/four

⇒ AO = a⎷3/2

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

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

The ratio of the diagonals is:

Ac/BD = a⎷three/a

= ⎷three/1

Therefore, AC : BD = ⎷3 : ane.