From a circle of radius 7 units, an arc length is cut by a chord of length 7 units. What is the arc length of the smaller portion (in units)?
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. .
The length of a chord (c) and the central angle subtended by the chord (θ) are related by:
c = 2rsin(θ/2)
⇒ 7 = 2(7) sin(θ/2)
⇒ sin(θ/2) = 1/2
From trigonometry:
θ/2 = 30º = π/6 radians
Thus, the central angle is:
θ = 2⋅π/6 = π/3 radians
The arc length (L) of a circle is given by:
⇒ L = r⋅θ
⇒ L = 7π/3 units
Since the chord divides the circle into two arcs, the smaller arc corresponds to the central angle π/3, while the larger arc corresponds to the complementary angle:
⇒ 2π − π/3 = 5π/3
The arc length of the smaller portion is therefore:
⇒ 7π/3 units