Aright triangle △abc with right angle c is inscribed in a circle. find the radius of this circle if:
b
given m∠c = 90°, k(o, r) inscribed in △abc, ac = 18 cm, m∠b = 30°. find r.

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

what’s the question?

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