In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord
Solution:
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
(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
☛ Check: NCERT Solutions Class 10 Maths Chapter 12
Video Solution:
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord
NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.2 Question 5
Summary:
The length of the arc APB, area of the sector AOBP and area of the segment of a circle of radius 21 cm in which an arc subtends an angle of 60° at the centre are 22 cm, 231 cm2 and (231 - 441√3/4) cm2 respectively.
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