Lengths of Diagonals of a Rhombus Abcd Are 16 Cm and 12 Cm. Find the Side and Perimeter of the Rhombus. - Mathematics

QUESTION

Lengths of diagonals of a rhombus ABCD are 16 cm and 12 cm. Find the side and perimeter of the rhombus.

SOLUTION

ABCD is a rhombus.

Here, segment AC and segment BD are the diagonals of the rhombus ABCD.

l(AC) = 16 cm and l(BD) = 12 cm.
Let the diagonals of rhombus ABCD intersect at point O.
l(AO) = 12 l(AC)
…[Diagonals of a rhombus bisect each other]
∴l(AO) = 12 × 16
∴l(AO) = 8 cm
Also, l(DO) = 12 l(BD)
…[Diagonals of a rhombus bisect each other]
∴l(DO) = 12 × 12
∴l(DO) = 6 cm
In ∆DOA,
m∠DOA = 90°
..[Diagonals of a rhombus are perpendicular to each other]
[l(AD)]² = [l(AO)]² + [l(DO)]²
…[Pythagoras theorem]

= (8)² + (6)²
= 64 + 36
∴[l(AD)]² = 100
∴l(AD) = √100
… [Taking square root of both sides]
∴l(AD) = 10 cm
∴l(AB) = l(BC) = l(CD) = l(AD) = 10 cm
…[Sides of a rhombus are congruent]
Perimeter of rhombus ABCD
= l(AB) + l(BC) + l(CD) + l(AD)
= 10+10+10+10
= 40 cm
∴The side and perimeter of the rhombus are 10 cm and 40 cm respectively.