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2
2
⇒ PQ = 36 + 64 = 100 = 10 \ PQ = 10 cm
Thus, the length of each side of the rhombus is 10 cm.
Rectangle

A parallelogram with equal angles and equal opposite sides A B
is called a rectangle.
In the adjoining figure, ABCD is a rectangle.
E
Here, AB = DC and AD = BC.

And ∠ A = ∠B = ∠C = ∠D D C
Properties of a Rectangle

• Opposite sides are parallel and equal.

• Opposite interior angles are equal, i.e., all rectangles are parallelograms, whose properties
apply to rectangles.
• Adjacent angles make a pair of supplementary angles.

• The diagonals of a rectangle are equal and bisect each other. That is AC = BD, and AE = EC =
DE = EB. The diagonals of a rectangle do not intersect at a right angle unless the rectangle is a
square.
• All the four angles are of 90°, i.e., ∠A = ∠B = ∠C = ∠D = 90˚.

Example 24: Prove that the diagonals of a rectangle are equal and bisect each other.
Solution: Given, ABCD is a rectangle with diagonals AC and BD intersecting at the point O.

We have to prove that, OA = OC, OB = OD and AC = BD
A B
From ∆ABC and ∆BAD,

AB = BA (Common) O

∠ABC = ∠BAD (Each equal to 90°)
D C
BC = AD (Opposite sides of a rectangle)
\ DABC ≅ DBAD (By SAS congruence)

⇒ AC = BD

Hence, the diagonals of the rectangle are equal.
Now, from DOAB and DOCD,
∠OAB = ∠OCD (Alternate angles)

∠OBA = ∠ODC (Alternate angles)

AB = CD (Opposite sides of a rectangle)
\ DOAB ≅ DOCD (By ASA congruence)

⇒ OA = OC and OB = OD.
Thus, the diagonals of a rectangle bisect each other.

Hence, the diagonals of a rectangle are equal and bisect each other.

73 Quadrilaterals