Test: Diagonals - GMAT MCQ

Given:

The length of a rectangular field is twice its breadth.

The area of the field is 288 sq.m.

Concept used:

Area of a rectangle = Length × Breadth

Calculation:

Let the length and breadth of the rectangular field be 2d and d meter respectively.

According to the concept,

2d × d = 288

⇒ 2d² = 288

⇒ d² = 144

⇒ d = ± 12

⇒ d = +12 (length can't be negative)

⇒ 2d = 24

∴ The length of the field is 24 meter.

Given:-

∠C = 60°  

Calculation:-

∵ ABCD is a rhombus, So, its all sides are equal.

∴ BC = DC 

⇒ ∠BDC = ∠DBC = x°   

In Δ BCD,

⇒ x° + x° + 60° = 180° 

⇒ 2x = 120

⇒ x = 60°

∵ ∠BDC = ∠DBC = ∠DCB = 60° 

∴ BD = BC = CD

Let BD = BC = CD = a,

In right angle Δ AOB,

AB² = OA² + OB²

OA² = AB² - OB²

OA² = a² - (a/2)² 

OA² = 3a²/4

OA = √3a/2

∵ AC = 2 × OA

AC = √3a

∴ AC : BD = √3a : a

AC : BD = √3 : 1 

Given:

Interior angle(I) – Exterior angle(E) = 144° 

Formula used:

For any regular polygon,

Interior angle(I) + Exterior angle(E) = 180° 

Each exterior angle = 360°/n, where

n = number of sides

Calculation:

ATQ,

⇒ I + E = 180° and I – E = 144°  

Adding both the eq.

⇒ 2I = 324 

⇒ I = 324/2 = 162° 

Now, putting the value,

⇒ E = 180° – 162° = 18°

Using the formula of E,

Number of sides = 360/18° = 20

∴ The number of sides of the regular polygon is 20.

Given:

External angle = 45° 

Formula used:

External angle = (360°/n)

Number of diagonal of a n side polygon = (n² - 3n)/2

Where, n = Equal to the number of side of a polygon

Calculation:

External angle = (360°/n)

⇒ 45° = (360°/n)

⇒ n = 8 

Now, Number of diagonal of a 'n' side polygon

⇒ (n² - 3n)/2

⇒ (64 - 24)/2

⇒ 20

∴ The number of diagonal is 20.

Figure:

Calculation:

As the diagonals of a rectangle intersect each other,

⇒ AO = OB

⇒ ∠OBA = ∠OAB = 25° [∵ Angle opposite to equal side are equal]

By angle sum property in ΔAOB,

⇒ ∠AOB + ∠OAB + ∠OBA = 180°

⇒ ∠AOB + 25° + 25° = 180°

⇒ ∠AOB = 130°

By linear pair property,

⇒ ∠DOA + ∠AOB = 180°

⇒ ∠DOA + 130° = 180°

⇒ ∠DOA = 50°

∴ Both diagonals make 50° angle with each other.

Side of the rhombus, a = 15 cm

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

As we know,

⇒ 225 = 144 + (d₂/2)²

⇒ (d₂/2)² = 81

⇒ d₂/2 = 9

⇒ d₂ = 18 cm

Given:

Sum of all interior angles = 2160° 

Formula used:

Sum of interior angles of polygon = (n - 2) × 180° 

Where 'n' is the number of sides of the polygon.

Calculation:

∵ The sum of all the angles of the polygon = 2160° 

⇒ (n - 2) × 180° = 2160° 

⇒ n - 2 = 2160°/180° 

⇒ n - 2 = 12

⇒ n = 12 + 2

⇒ n = 14

Concept:

Each angle of n-sided polygon = ((n - 2) × 180)/n

Number of diagonals of n - sided polygon = n(n - 3)/2

Calculation:

Each interior angle = 150° 

150° = (n - 2) × 180)/n

⇒ 6n - 12 = 5n

n = 12 = Total side of the polygon

∴ Number of diagonals of n - sided polygon = n(n - 3)/2 = 108/2

∴ Number of diagonals of n - sided polygon = 54

Area of the Trapezium = 1/2 × (Sum of the parallel sides) × (Distance between parallel sides)

⇒ 1/2 × (53 + 68) × 16

⇒ 1/2 × 121 × 16

∴ Area of the Trapezium = 968 cm²

Each interior angle of a regular polygon is 135,

⇒ Exterior angle = 180° - Interior angle = 45°

⇒ Number of sides of polygon = 360°/Exterior angle = 8

∴ Number of diagonals = n(n - 3)/2 = 8 × (8 - 3)/2 = 20, where n is the number of sides of a polygon.