7. (a) Angle POQ: Since PQ is tangent to the circle at point P, the angle between the tangent and the radius at the point of contact is always a right angle. So, angle POQ is a right angle, which is π/2 radians.
b) Area of sector POR: The area of a sector of a circle is given by the formula: Area of sector = (angle/2π) × πr². Here, angle POR is 90 degrees or π/2 radians, and the radius r is 12 cm. Thus, the area of sector POR is: Area of sector = (π/2)/(2π) × π(12)² = 1/4 × 144π = 36π cm².
c) Area of the shaded region: Since PQ is tangent to the circle, triangle POQ is a right triangle. The area of the shaded region consists of the area of sector POR minus the area of triangle POQ. The area of triangle POQ can be found using the formula for the area of a triangle: Area of triangle = 1/2 × base × height. Here, the base is PQ and the height is OQ. Since OP = OQ (radii of the same circle), triangle POQ is an isosceles right triangle, so OQ = OP = 12 cm. So, the area of triangle POQ is: Area of triangle = 1/2 × PQ × OQ = 1/2 × 12 × 12 = 72 cm². Therefore, the area of the shaded region is: Area of sector POR - Area of triangle POQ = 36π - 72.
8. (a) To find the angle AOB in radians, we can use the formula for the area of a sector:
Area of sector = (angle / 2π) × πr²
Given that the area of sector AOB is 32 cm² and the radius is 8 cm, we have:
32 = (angle / 2π) × π × 8²
32 = (angle / 2) × 64
angle = (32 × 2) / 64 = 1 radian
b) To find the area of triangle AOC, we can use the formula for the area of a triangle:
Area of triangle = (1/2) × base × height
In triangle AOC, AC is the height and OC is the base. Since AC is a tangent to the circle at point A, it is perpendicular to OC. So, triangle AOC is a right triangle.
Base OC = 8 cm (radius of the circle)
Height AC = 8 cm (tangent to the circle)
Area of triangle AOC = (1/2) × 8 × 8 = 32 cm²
c) The shaded region is formed by the sector AOB and triangle AOC. We've already found the area of the sector (32 cm²) and the area of triangle AOC (32 cm²). Therefore, the area of the shaded region is the difference between these two areas:
Area of shaded region = Area of sector AOB - Area of triangle AOC
= 32 cm² - 32 cm² = 0 cm²
So, the area of the shaded region is 0 cm².
