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Some Special Parallelograms

Parallelograms with special properties are given special names. Rhombus, rectangle and square
are some special parallelograms.
Rhombus

If a parallelogram has all the four sides equal, then it is called a rhombus. A B

In the adjoining figure, ABCD is a rhombus. Here, AB = BC = CD = AD.
Properties of a Rhombus E

• A rhombus has all the properties of a parallelogram and a kite.
D C
• Opposite angles of a rhombus are equal; that is, ∠A = ∠C and ∠B = ∠D.
• The special property of a rhombus is that its diagonals are perpendicular bisector to each
other, i.e., it makes an angle of 90° at the point of intersection of the diagonals and AE = EC and
DE = EB.
Example 22: Prove that the diagonals of a rhombus bisect each other at right angles.

Solution: Let us take a rhombus WXYZ. W X
Since, a rhombus is also a parallelogram, so its diagonals bisect each other.

So, WO = OY and ZO = OX. O
In DWOZ and DWOX, we have

WZ = WX (All sides of a rhombus are equal) Z Y
WO = WO (Common)
ZO = OX (Diagonals of a ||gm bisect each other)

\ DWOZ ≅ DWOX (By SSS criterion of congruence)
Now, ∠WOZ = ∠WOX (Corresponding parts of congruent triangles are equal)

We have, ∠WOZ + ∠WOX = 180° (Linear pair)
⇒ ∠WOZ + ∠WOZ = 180° activity

⇒ 2∠WOZ = 180° l Take two sticks of any
⇒ ∠WOZ = 90° equal lengths (suppose each of length 10 cm).
l Paste them on a cardboard sheet as they are
Hence, the diagonals of a rhombus bisect each other crossing each other such that they bisect each
at right angles. other at right angles.

Example 23: The diagonals of a rhombus are 16 cm What shape will be formed by joining their end
and 12 cm. Find the length of a side. points? Give reason.
Solution: Let PQRS be the rhombus whose diagonals PR = 12 cm and QS = 16 cm.

Since the diagonals of a rhombus bisect each other at right angles. P S
 1   1  6 cm 8 cm
PO =   2 × 12 cm = 6 cm and QO =   2 × 16 cm = 8 cm O




In right-angled triangle POQ, we have 8 cm 6 cm
2
2
2
PQ = PO + QO (Using Pythagoras property) Q R
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