The area of the shaded region, formed by folding a circle with a 24 cm diameter along a vertical chord AB, is 192π - 144√3 square centimeters. The answer is (d) 192π - 144√3.
To find the area of the shaded region, consider a circle with a diameter of 24 cm folded along a vertical chord AB, coinciding point D with center C.
After folding, a sector and a triangle are formed. The sector is the larger shaded region, and the triangle is the smaller one. The area of the sector is calculated using the formula:
Area of Sector = (Central Angle / 360) * π * r^2
With a central angle of 180° (since it's folded along a diameter) and a radius (r) of 12 cm, the sector's area is (180/360) * π * 12^2.
Next, the area of the triangle formed by the folded chord AB is found. The base is the diameter (24 cm), and the height is the radius (12 cm), using the formula:
Area of Triangle = (1/2) * Base * Height
Subtract the triangle's area from the sector's area to get the total shaded area:
Shaded Area = Area of Sector - Area of Triangle
After simplifying the expression, the final answer is:
(d) 192π - 144√3