Answer:
The area of the shaded region = (36·π - 72) cm²
The perimeter of the shaded region = (6·π + 12·√2) cm
Step-by-step explanation:
The given figure is a sector of a circle and a segment of the circle is shaded
We have that since the arc AC subtends an angle 90° at the center of the circle, the sector is a quarter of a circle, which gives;
Area of sector = 1/4×π×r²
As seen the radius, r = AB = 12 cm
∴ Area of sector = 1/4×π×12² = 36·π cm²
The area of the segment AB = Area of sector ABC - Area of ΔABC
Area of ΔABC = 1/2×Base ×Height =
Since the base and the height = The radius of the circle = 12 cm, we have;
Area of ΔABC = 1/2×12×12 = 72 cm²
The area of the segment AB = 36·π cm² - 72 cm² = (36·π - 72) cm²
The area of the shaded region = The area of the segment AB = (36·π - 72) cm²
The perimeter of the shaded region = 1/4 perimeter of the circle with radius r + Line Segment AC
The perimeter of the shaded region = 1/4 × π × 2 × r + √(12² + 12²) = 1/4 × π × 2 × 12 + 12·√2 = (6·π + 12·√2) cm
