Question

in a circle of diameter 21 cm an arc subtends an angle of 60 dregee at the centre then the lenght of the arc is​

Answer 1

Answer:

In the circle with radius r and the angle at the centre with a degree measure of θ,

(i) Length of the Arc = θ/360° × 2πr

(ii) Area of the sector = θ/360° × πr2

(iii) Area of the segment = Area of the sector - Area of the corresponding triangle

Let's draw a figure to visualize the problem.

Here, r = 21 cm, θ = 60°

Visually it’s clear from the figure that,

Area of the segment APB = Area of sector AOPB - Area of ΔAOB

Step-by-step explanation:

(i) Length of the Arc, APB = θ/360° × 2πr

= 60°/360° × 2 × 22/7 × 21 cm

= 22 cm

(ii) Area of the sector, AOBP = θ/360° x πr2

= 60°/360° × 22/7 × 21 × 21 cm2

= 231 cm2

(iii) Area of the segment = Area of the sector AOBP - Area of the triangle AOB

To find the area of the segment, we need to find the area of ΔAOB

In ΔAOB, draw OM ⊥ AB.

Consider ΔOAM and ΔOMB,

OA = OB (radii of the circle)

OM = OM (common side)

∠OMA = ∠OMB = 90° (Since OM ⊥ AB)

Therefore, ΔOMB ≅ ΔOMA (By RHS Congruency)

So, AM = MB (Corresponding parts of the congruent triangles are always equal)

∠AOM = ∠BOM = 1/2 × 60° = 30°

In ΔAOM,

cos 30° = OM/OA and sin 30° = AM/OA

√3/2 = OM/r and 1/2 = AM/r

OM = (√3/2) r and AM = (1/2) r

AB = 2AM

AB = 2 × (1/2) r

AB = r

Therefore, area of ΔAOB = 1/2 × AB × OM

= 1/2 × r × (√3/2) r

= 1/2 × 21 cm × (√3/2) × 21 cm

= 441√3/4 cm2

Area of the segment formed by the chord = Area of the sector AOBP - Area of the triangle AOB

= (231 - 441√3/4) cm2

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