Question

A sector of a circle has an area of 32πcm^2 and contains a central angle of 210 degrees.calculate the radius of the circle and length of the cord that subtends the angle given above

Answer 1

[AI-generated answer — Bot GPT 3.5]

The area of a sector of a circle is given by the formula:

Area of sector = (θ/360) * π * r^2

Where θ is the central angle in degrees and r is the radius of the circle.

Given that the area of the sector is 32πcm^2 and the central angle is 210 degrees, we can plug these values into the formula and solve for the radius:

32π = (210/360) * π * r^2
32 = (7/12) * r^2

r^2 = 32 / (7/12)
r^2 = 32 * (12/7)
r^2 = 384/7
r ≈ 7.02 cm

Therefore, the radius of the circle is approximately 7.02 cm.

To calculate the length of the chord that subtends the central angle of 210 degrees, we can use the formula:

Chord length = 2 * r * sin(θ/2)

Where r is the radius of the circle and θ is the central angle in radians.

Converting 210 degrees to radians:

210 degrees * (π/180) = 7π/6 radians

Now plug in the values to get:

Chord length = 2 * 7.02 * sin(7π/6)
Chord length = 14.04 * sin(7π/6)
Chord length = 14.04 * (-1/2)
Chord length = -7.02

Therefore, the length of the chord that subtends the central angle of 210 degrees is 7.02 cm.