The value of angle ABD is 270°.
Step 1: Calculate the radius of the circle:
We are given the area of the circle, which is 56.7 cm². We can use the formula for the area of a circle to calculate the radius:
Area = πr²
56.7 = πr²
r = √(56.7 / π) = 4.5 cm
Step 2: Calculate the angle measure of sector AOB:
We are given that AD = 4.5 cm, which is equal to the radius of the circle. This means that triangle AOB is a right isosceles triangle. Therefore, we can calculate the angle measure of sector AOB as follows:
Angle AOB = 2 * ∠OAB
∠OAB = 90° - 45° = 45°
Angle AOB = 2 * 45° = 90°
Step 3: Calculate the ratio of sector AOB to the whole circle:
The ratio of the angle measure of sector AOB to the angle measure of the whole circle (360°) is equal to the ratio of the area of sector AOB to the area of the whole circle:
Ratio = Angle AOB / 360° = Area of sector AOB / Area of the whole circle
Ratio = 90° / 360° = Area of sector AOB / 56.7 cm²
Step 4: Calculate the area of sector AOB:
Solving for the area of sector AOB:
Area of sector AOB = Ratio * Area of the whole circle
Area of sector AOB = (90° / 360°) * 56.7 cm²
Area of sector AOB = 13.675 cm²
Step 5: Calculate the angle measure of sector ABD:
Since angle AOB = 90° and the area of sector AOB is 13.675 cm², then the area of sector ABD is 56.7 cm² - 13.675 cm² = 43.025 cm².
Finally, we can calculate the angle measure of sector ABD:
Angle ABD = (Area of sector ABD / Area of the whole circle) * 360°
Angle ABD = (43.025 cm² / 56.7 cm²) * 360°
Angle ABD = 270°
Therefore, the value of angle ABD is 270°.