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

Given:
Radius (r) = 21 cm
Angle subtended by the arc at the center (θ) = 60°

(i) Length of the arc:
The length of an arc (l) is given by the formula:
l = (θ/360°) × 2πr
Substituting the given values:
l = (60°/360°) × 2π(21 cm)
l = (1/6) × 42π cm
l = 7π cm
Therefore, the length of the arc is 7π cm, which is approximately 21.99 cm.

(ii) Area of the sector:
The area of a sector (A_sector) is given by the formula:
A_sector = (θ/360°) × πr²
Substituting the given values:
A_sector = (60°/360°) × π(21 cm)²
A_sector = (1/6) × 441π cm²
A_sector = 73.5π cm²
Therefore, the area of the sector is 73.5π cm², which is approximately 230.91 cm².

(iii) Area of the segment:
The area of the segment (A_segment) is the difference between the area of the sector and the area of the triangle formed by the two radii and the chord.
First, let's find the area of the triangle:
Area of the triangle (A_triangle) = (1/2)r²sinθ
A_triangle = (1/2)(21 cm)²sin60°
A_triangle = (1/2)(441 cm²)(√3/2)
A_triangle = (441√3)/4 cm²
A_triangle ≈ 190.96 cm²
Now, let's find the area of the segment:
A_segment = A_sector - A_triangle
A_segment = 73.5π cm² - (441√3)/4 cm²
A_segment ≈ 230.91 cm² - 190.96 cm²
A_segment ≈ 39.95 cm²
Therefore, the area of the segment is approximately 39.95 cm².