Rhombus MCQ Quiz - Objective Question with Answer for Rhombus - Download Free PDF

Given:

Diagonals of the rhombus, d₁ = 47 cm and d₂ = 48 cm

Formula used:

Area of a rhombus = (1/2) × d₁ × d₂

Calculation:

Area = (1/2) × 47 cm × 48 cm

⇒ Area = (1/2) × 2256 cm²

⇒ Area = 1128 cm²

∴ The correct answer is option (4).

Given:

The length of side of a rhombus is 5 m and one of its diagonal is 8 m.

d1 = 8 m

Side = 5 m

Formula used:

In a rhombus, the diagonals bisect each other at right angles.

If d₁ and d₂ are the lengths of the diagonals, then:

d₁² + d₂² = 4 × (side)²

Calculation:

Let one diagonal be 8 m (d₁) and the other diagonal be d₂.

⇒ d₁² + d₂² = 4 × (5)²

⇒ 8² + d₂² = 4 × 25

⇒ 64 + d₂² = 100

⇒ d₂² = 100 - 64

⇒ d₂² = 36

⇒ d₂ = 6 m

∴ The correct answer is option 3.

Given:

The lengths of diagonals of a rhombus are 24 cm and 32 cm respectively.

Formula Used:

Area of a rhombus = (1/2) × d₁ × d₂

Altitude = Area / Side length

Calculation:

Diagonal 1 (d₁) = 24 cm

Diagonal 2 (d₂) = 32 cm

Side length of rhombus:

Each side of the rhombus forms a right triangle with half of each diagonal:

Side length = \(\sqrt{(d_1/2)^2 + (d_2/2)^2}\)

Side length = \(\sqrt{(24/2)^2 + (32/2)^2}\)

Side length = \(\sqrt{12^2 + 16^2}\)

Side length = \(\sqrt{144 + 256}\)

Side length = \(\sqrt{400}\)

Side length = 20 cm

Area of rhombus:

Area = (1/2) × 24 × 32

Area = 12 × 32

Area = 384 cm²

Altitude of rhombus:

Altitude = Area / Side length

Altitude = 384 / 20

Altitude = 19.2 cm

The altitude of the rhombus is 19.2 cm.

Given:

Length of one diagonal (d₁) = 12 cm

Area of the rhombus (A) = 108 cm²

Formula used:

Area of a rhombus = \(\dfrac{1}{2} \times d_1 \times d_2\)

Calculation:

108 = \(\dfrac{1}{2} \times 12 \times d_2\)

⇒ 108 = 6 × d₂

⇒ d₂ = \(\dfrac{108}{6}\)

⇒ d₂ = 18 cm

∴ The correct answer is option (1).

Given:

Length of one diagonal (d₁) = 12 cm.

Area of the rhombus (A) = 108 cm².

Formula Used:

Area of rhombus = (1/2) × d₁ × d₂

Calculation:

We know the area of the rhombus:

108 = (1/2) × 12 × d₂

⇒ 108 = 6 × d₂

⇒ d₂ = 108 / 6

⇒ d₂ = 18 cm

The length of the second diagonal of the rhombus is 18 cm.

Area of rhombus = Product of both diagonals/ 2,

⇒ 840 = P × Q /2,

⇒ P × Q = 1680,

Using Pythagorean Theorem we get,

⇒ (P/2)² + (Q/2)² = 37²

⇒ P2 + Q2 = 1369 × 4

⇒ P2 + Q2 = 5476

Using perfect square formula we get,

⇒ (P + Q)² = P² + 2PQ + Q²

⇒ (P + Q)² = 5476 + 2 × 1680

⇒ P + Q = 94

Hence option 4 is correct.

Given : 

Area of Rhombus = 5544 m²

One of its diagonal = 72 m

Formula Used : 

Area of Rhombus = 1/2 × D₁ × D₂

Calculation : 

According to question,

⇒ 5544 = 1/2 × 72 × D₂

⇒ D₂ = 154

Now,  D₂/2 = 77, D₁/2 = 36

Now by using Pythagoras theorem,

⇒ AD² = 77² + 36² = 1296 + 5929 = 7225

⇒ AD = √7225 = 85 m 

Perimeter = 4 × 85 = 340 m

∴ The correct answer is 340 m.

Given:

Perimeter of Rhombus = 148 cm

One diagonal = 24 cm

Formula used:

Perimeter of Rhombus = 4 × side

Area of Rhombus = 1/2 × d1 × d2

where, d1 and d2 are diagonals of rhombus

Calculation:

Perimeter = 4 × side

⇒ 148 = 4 × side

⇒ side = 37 cm

In right angled triangle  ΔAOB,

⇒ AB² = AO² + OB²

⇒ (37)² = (12)² + OB²

⇒ 1369 = 144 + OB²

⇒ OB² = (1369 – 144)

⇒ OB² = 1225 cm²

⇒ OB = 35 cm

BD = 2 × OB

⇒ 2 × 35 cm

⇒ 70 cm

Area of Rhombus = (1/2 × 24 × 70) cm²

⇒ 840 cm²

∴ Area of Rhombus is 840 cm²

Calculation:

Side of the rhombus, a = 15 cm

Hence, length of the diagonal of the rhombus = 15 × [160/100] = 24 cm

As we know,

\(\Rightarrow {a^2}{\rm{}} = {\rm{}}\frac{{d_1^2}}{4}{\rm{}} + {\rm{}}\frac{{d_2^2}}{4}\)

\(\Rightarrow 15^2{\rm{}} = {\rm{}}{\left( {\frac{{24}}{2}} \right)^{2{\rm{\;}}}}{\rm{}} + {\rm{}}{\left( {\frac{{d2}}{2}} \right)^2}\)

⇒ 225 = 144 + (d₂/2)²

⇒ (d₂/2)² = 81

⇒ d₂/2 = 9

Given:

Perimeter of rhombus = 120 m

Calculation:

Length of each side of rhombus = 120/4 = 30 m

Height of rhombus = 15m

Area of rhombus = base of length × height

= 30 × 15

= 450 sq.m

∴ Area of rhombus is 450 m²

Area of Rhombus = 1/2 × product of diagonals

⇒ 720 = 1/2 × product of diagonals

⇒ Product of diagonals = 1440

(Side of rhombus)² = (half of one diagonal)² + (half of the other diagonal)²

⇒ (One diagonal)² + (Other diagonal)² = 41 × 41 × 4

(Sum of two diagonal)² = (One diagonal)² + (other diagonal)² + 2 × product of diagonals

⇒ (Sum of two diagonal)² = 6724 + 2880 = 9604

Given:

Side (a) of rhombus = 13 cm 

Length of one diagonal (p) = 24 cm 

Formulas used:

a = (p² + q² )^1/2÷ 2

Area of rhombus = 1/2 × p × q

Calculation:

Let the other diagonal of rhombus = q 

⇒ (24² + q²)^1/2 ÷ 2 = 13

⇒ (576 + q²)^1/2 = 26

⇒ 576 + q² = (26)²

⇒ 676 - 576 = q²

⇒ q = √100 = 10 

Area = 1/2 × 24 × 10 = 120 cm²

∴ The correct answer is 120 cm².

Given:

Perimeter of the rhombus = 100 cm

Diagonal, D₁ = 14 cm

Concept used:

All sides of a rhombus are equal.

In a  rhombus, diagonals bisect each other perpendicularly.

Pythagoras' Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Formula: 

Area of a rhombus = ½ × D1  × D2

Where, D1 and D2 are the diagonals of rhombus.

Perimeter of rhombus = 4 × Length of each side   

Calculation:

Let, second diagonal = 2y

Each side of the rhombus = 100/4 cm = 25 cm

Taking a triangular segment of rhombus and applying Pythagoras theorem,

⇒ 252 – 72 = y2

⇒ y2 = 625 – 49

⇒ y = 24 cm

So, Length of second diagonal = 2 × 24 = 48 cm

Area of the rhombus

⇒ ½ × 14 × 48 cm²

⇒ 336 cm2 

∴ The area of the rhombus is 336 cm²

Diagonals of a Rhombus are perpendicular bisector

Let ABCD be a rhombus and AC = 6 cm with midpoint O and Side AB = 6 cm

So, in ΔAOB,

⇒ AO² + OB² = AB²

⇒ (6/2)² + OB² = 6²

⇒ 9 + OB² = 36

⇒ OB² = 27

⇒ OB = 3√3 cm

⇒ BD = 2 × OB = 6√3 cm

⇒ Area of Rhombus = (1/2) × (Product of diagonal of Rhombus)

Given:

D₁ =  8√3 cm , D₂ = side of the rhombus = a 

Concept:

The diagonals of a rhombus bisect each other at right angles.

Formula used:

Area of a rhombus = 1/2 × Diagonal₁ × Diagonal₂

Calculation:

By Pythagoras theorem

a² = a²/4 + (4√3)²

⇒ a² = 64 

⇒ a = 8 cm = D₂

∴ The area of the rhombus = 1/2 × 8 × 8√3

⇒ 32√3 sq. cm