Numerical Ability Test - 27

Given:

The length of the chord is 16 cm and the radius is 10 cm.

Concept Used:

The radius of a circle bisects the chord of the circle perpendicularly.

Formula used:

In the right-angle triangle, by Pythagoras theorem

(Hypotenuse)² = (Perpendicular)² + (Base)²

Calculation:

Let the two chords be AB = 16 cm

As, the radius of the circle bisect perpendicularly,

AL = BL = 16/2 = 8 cm

In Δ AOL, ∠ALO = 90°

⇒ (AO)² = (OL)² + (AL)²

⇒ 10² = (OL)² + (8)²

⇒ (OL)² = 100 - 64 = 36

⇒ OL = 6 cm

∴ The distance of the chord from the centre of the circle is 6 cm.

Given:

Point P = (-2, 3)

Equation of the given line: x + 2y - 3 = 0

Formula used:

Slope of a line ax + by + c = 0 is given by: 

Slope of the perpendicular line:

⇒ m = 2

Using point P(-2, 3) in the equation y - y₁ = m(x - x₁):

⇒ y - 3 = 2(x + 2)

⇒ y - 3 = 2x + 4

⇒ y - 2x = 7

⇒ y - 2x - 7 = 0

∴ The correct answer is option (4).

Given:

What is the measure of each exterior angle of a regular octagon?

Formula used:

Each exterior angle of a regular polygon

Where, n = number of sides

Calculation:

n = 8 (since an octagon has 8 sides)

Given:

Vertices of the quadrilateral are (3, 0), (4, 5), (-1, 4), and (-2, -1).

Formula Used:

Area of a quadrilateral with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄) is given by:

Area = (1/2) | x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ - (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁) |

Calculation:

Substitute the coordinates into the formula:

Vertices: (3, 0), (4, 5), (-1, 4), (-2, -1)

Area = (1/2) | 3×5 + 4×4 + (-1)×(-1) + (-2)×0 - (0×4 + 5×(-1) + 4×(-2) + (-1)×3) |

⇒ Area = (1/2) | 15 + 16 + 1 + 0 - (0 - 5 - 8 - 3) |

⇒ Area = (1/2) | 32 - (-16) |

⇒ Area = (1/2) | 32 + 16 |

⇒ Area = (1/2) | 48 |

⇒ Area = 24

The area of the quadrilateral is 24 sq. units.

Given:

Radius (r) = 28 cm

Central Angle (θ) = 45°

Formula used:

⇒ Area = 308 cm²

∴ The correct answer is option (1).

Given:

(A) All the congruent figures are similar

(B) All the similar figures are congruent

Formula Used:

To determine if the statements about congruent and similar figures are true.

Calculation:

Statement (A): All the congruent figures are similar.

Congruent figures have the same shape and size. Similar figures have the same shape but may differ in size. Since congruent figures have the same shape, they are also similar.

⇒ Statement (A) is true.

Statement (B): All the similar figures are congruent.

Similar figures have the same shape but can differ in size. Therefore, not all similar figures are congruent.

⇒ Statement (B) is false.

The correct answer is option 1 (Only (A)).

Given:

Side of an equilateral triangle = 24 cm.

Formula Used:

Radius of the incircle (r) of an equilateral triangle = (side × √3) / 6

Calculation:

Side of the equilateral triangle = 24 cm

Radius of the incircle (r) = (24 × √3) / 6

⇒ r = 4√3 cm

The radius of the incircle of the equilateral triangle is 4√3 cm.

Given:

The angle subtended by arc ABC at the center of the circle = 138°.

Chord AB is produced to a point P, and we need to find ∠CBP.

Formula used:

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circle's circumference.

Calculation:

Let ∠AQC be the angle subtended by arc ABC at the circle's circumference.

⇒ ∠AOC = 2 × ∠AQC

⇒ ∠AQC = 1/2 × ∠AOC = 1/2 × 138° = 69°

Now, in the cyclic quadrilateral ABQC, the exterior angle ∠CBP is equal to the interior opposite angle ∠AQC.

⇒ ∠CBP = ∠AQC = 69°

∴ The measure of ∠CBP is 69°.

Calculation

By the theorem,

LQ × LP = LY × LX

Let the length of XY be x.

⇒ 5 × 15 = 3 × (3 + x)

⇒ 25 = x + 3

⇒ x = 22

The length of XY is 22.

Formula used:

Direct common tangent = √ (distance between the two centers² - ( r₁  - r₂)²

Calculation:

Distance between the two centers is = d cm

As per the formula,

12 = √ (d2 - ( 8  - 3)².

⇒ 144 = d2 - 25

⇒ d² = 169

⇒ d = 13

∴ The correct option is 3