Question

Consider a ˚le with center O, and points B and C lying on its ˚umference. If the arc measure of arc BDC is 60 degrees, find the area of the region bounded by segment BC and arc BDC in terms of x.

a) (1/6)x^2π
b) (1/3)x^2π
c) (1/2)x^2π
d) (2/3)x^2π

Answer 1

The area of the region bounded by segment BC and arc BDC with a 60-degree arc on a circle with radius x is (1/6)x^2π, which is one-sixth of the circle's area.

Step-by-step explanation:

The question asks us to find the area of a region bounded by a chord and an arc on a circle. When we have an arc measure of 60 degrees on the circle with radius x, we can calculate the area of the sector (the 'slice' of the circle) formed by this arc. Since the entire circumference of the circle corresponds to an angle of 360 degrees and a complete circle's area is πx^2, the area of the sector corresponding to a 60-degree angle is one-sixth of the full circle's area, which is (1/6)x^2π.

Thus, the area of the region bounded by segment BC and arc BDC (assuming BC is the chord of the sector), would also approximate the area of the sector because over small angles, the chord and arc lengths are nearly equal, leading to a negligible difference between the area of the sector and the bounded region. Therefore, the correct answer is option (a) (1/6)x^2π.