Answer:
21.75 square units
Step-by-step explanation:
Draw a radial line from O to the point where AB intersects the circle. We'll call this point P.
Draw another radial line from O to the point where arc BC intersects the circle. We'll call this point Q.
OQ is equal to the radius of the circle, 2. And AQ is equal to the radius of the sector, 8. Therefore, the length of AO is 6.
Next, OP is equal to the radius of the circle, 2. Since AB is tangent to the circle, it is perpendicular to OP.
So we have a right triangle with a hypotenuse of 6 and a short leg of 2. Finding the angle ∠PAO:
sin ∠PAO = 2/6
∠PAO = asin(1/3)
That means the angle ∠BAC is double that:
∠BAC = 2 asin(1/3)
∠BAC ≈ 38.94°
Therefore, the area of the sector is:
A = (θ/360) πr²
A = (38.94/360) π(8)²
A ≈ 21.75
The area of the sector is approximately 21.75 square units.
