Final answer:
To find the area of the minor segment of a circle, calculate the area of the sector and subtract the area of the isosceles triangle formed by the radii and the chord.
For a circle with radius 14 cm and a 60-degree angle, the approximate area of the minor segment is 17.83 cm^2, which does not match the provided options.
Step-by-step explanation:
To find the area of the minor segment of a circle with a given radius and a central angle, we need to calculate the area of the circular sector and then subtract the area of the corresponding isosceles triangle.
First, calculate the area of the sector formed by the 60-degree central angle in a circle with a radius of 14 cm. The formula for the area of a sector is A = (\theta / 360) \times \pi \times r^2, where \(\theta\) is the central angle in degrees and \(r\) is the radius.
For a 60-degree angle, the area of the sector is:
A_sector = (60/360) \times \pi \times (14)^2 = (1/6) \times \pi \times 196 cm^2 approximately 102.67 cm^2.
Next, calculate the area of the isosceles triangle formed by the two radii and the chord. Using the formula A = 1/2 \times base \times height with the base equal to the length of the chord.
Since the triangle is isosceles and the central angle is 60 degrees, it forms an equilateral triangle, so the height can be found using Pythagoras' theorem or knowing that in an equilateral triangle, the height is \(\sqrt{3}/2\) times the side length.
The length of each side (the radius) is 14 cm, so the height of the triangle is h = 14 \times \sqrt{3}/2 approximately 12.12 cm.
Hence, the area of the triangle is A_triangle = 1/2 \times 14 \times 12.12 cm^2 approximately 84.84 cm^2.
Now subtract the area of the triangle from the area of the sector:
A_minor_segment = A_sector - A_triangle = 102.67 cm^2 - 84.84 cm^2 approximately 17.83 cm^2.
As the calculated area does not match any of the provided options (a. 22 cm^2, b. 33 cm^2, c. 44 cm^2, d. 66 cm^2), there may be an error in the given options or in the calculation. Please check the question and options or consult additional resources as necessary.