Class-8 Rhombus
Introduction to Rhombus
A parallelogram having all sides equal is known as rhombus. In the below given figure ABCD is a parallelogram having all sides equal i.e. AB = BC = CD = AD, hence we can conclude that ABCD is a rhombus.
Properties of Rhombus
- Opposite sides are parallel i.e., AB || CD and BC || AD
- All sides are equal i.e., AB = BC = CD = AD
- Opposite angles are equal i.e., ∠A = ∠C and ∠B = ∠D
- Diagonals bisect each other i.e., AO = OC and BO = OD
- Diagonals of a rhombus bisect each other at right angles i.e., ∠AOD = ∠COD = ∠AOB = ∠BOC = 90°
Example 1. Find the value of a, b and c in the below given rhombus.
Solution. Since all sides of a rhombus are equal.
AB = BC = CD = AD = 5 cm
=> c = 5 cm
Diagonals of a rhombus bisects each other.
AO = CO = 4 cm
a = 4 cm
DO = BO = 3 cm
b = 3 cm
Hence, a = 4 cm, b = 3 cm and c = 5 cm.
Example 2. In the below given figure, ABCD is a rhombus. If ∠A = 60°, find ∠CBD.
Solution. Opposite angles of a rhombus are same.
∠A = ∠C = 60°
∠B = 180° − 60° (Adjacent angles of a rhombus are supplementary)
= 120°
∠CBD + ∠BDC + 60° = 180°
=> p° + p° + 60° = 180°
=> 2p° = 180° − 60°
=> 2p° = 120°
=> p° = 120°⁄2
=> p° = 60°
Hence, ∠CBD = 60°.
Example 3. ABCD is a rhombus as shown in below given figure. Find the measure of ∠A.
Solution. ABCD is a rhombus and BCD is an isosceles triangle as BC = CD.
∠CBD = ∠BDC = 60°
∠CBD + ∠BDC + ∠BCD = 180° (Sum of all angles of a triangle = 180°)
=> 60° + 60° + ∠BCD = 180°
=> 120° + ∠BCD = 180°
=> ∠BCD = 180° − 120°
=> ∠BCD = 60°
=> ∠C = 60°
Opposite angles of a rhombus are equal
∠A = ∠C = 60°
Hence, ∠A = 60°.
Example 4. In the below given figure, ABCD is a rhombus. If ∠OCD = 30°, then find the measure of ∠OBC.
Solution. ∠BAC = ∠ACD (AC is traversing through two parallel lines AB and CD)
∠BAC = 30°
Here ABC is an isosceles triangle where AB = BC = CD = AD
So, ∠BAC = ∠BCA = 30°
Here, OBC is a right-angle triangle.
∠OBC + ∠BOC + ∠BCO = 180° (Sum of all angles of a triangle = 180°)
=> ∠OBC + 90° + 30° = 180°
=> ∠OBC + 120° = 180°
=> ∠OBC = 180° − 120°
=> ∠OBC = 60°
Hence, ∠OBC = 60°.
Example 5. If the diagonals of a rhombus are 12 cm and 16 cm, find the length of each side.
Solution.Here ABCD is a rhombus, where AC = 16 cm and BD = 12 cm.
AO = CO = 8 cm (As we know rhombus diagonals bisects each other)
BO = DO = 6 cm
Rhombus diagonals bisect each other at an angle 90°. So, AOB is a right-angle triangle.
AB2 = AO2 + BO2 (As per Pythagoras theorem)
=> AB2 = 82 + 62
=> AB2 = 64 + 36
=> AB2 = 100
=> AB = √ 100
=> AB = 10 cm
Length of rhombus sides is equal to 10 cm.
Class-8 Rhombus MCQ
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