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.
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,
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
⇒ 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