Answer:
The radius = 18 cm
Step-by-step explanation:
* Lets take about the inscribed triangle in a circle
- If the three vertices of a triangle lie on the circumference of a circle,
then this triangle is inscribed in the circle
- The vertices of the triangle are inscribed angles in the circle
- The inscribed angle opposite to a circle's diameter is always a
right angle (its measure is 90°)
- Now lets solve the problem
∵ Δ ABC is a right triangle at C
∴ m∠C = 90°
∵ Δ ABC is inscribed in a circle
∴ A , B , C lie on the circumference of the circle
∴ ∠A , ∠B , ∠C are inscribed angles in the circle
∴ m∠C = 90°
∵ ∠C is opposite to the side AB
∴ AB is the diameter of the circle ⇒ from the bold note above
∵ m∠B = 30°
∵ AC = 18 cm
- Lets use the trigonometry function to find the length of AB
* In Δ ABC
∵ AC is opposite to angle B
∵ AB is the hypotenuse
∵ sin Ф = opposite/hypotenuse
∴ sin B = AC/AB
∴ sin (30)° = 18/AB ⇒ using cross multiplication
∴ AB sin (30)° = 18 ⇒ divide both sides by sin (30)°
∴ AB = 18/sin(30)°
∵ sin(30)° = 1/2
∴ AB = 18/(1/2) = 36 cm
∵ AB is the diameter of the circle
∵ The length of the radius of a circle = 1/2 the length of the diameter
∴ The radius = 1/2 × 36 = 18 cm
