How to find the radius of a circle?

Question:

In the figure shown, triangle ABC inscribed in the circle O. If AB = 32, the angle A = 80^o, AC = 45. What is the radius of the circle?

Solution:

When a triangle inscribed in a circle, then 2 r = a/sin A = b/sin B = c/sin C.

In which, r is the radius of the circle, a, b and c are three sides of the triangle ABC. In which, a is the side opposite the angle A, b is the side opposite the angle B, and c is the side opposite the angle C. So, a = BC which is unknown, b = AC = 45, and c = AB = 32.

Apply the Cosine law to the triangle ABC,

a² = b² + c² - 2 b c cos A

= AC² + AB² - 2 × AC × AB × cos80^o

= 45² + 32² - 2 × 45 × 32 × 0.1736

= 2549.03

a = 50.5

using formula,

2 r = a / sin A

r = a / (2 sin A)

= 50.5 / (2 × sin 80^o)

= 50.5 / (2 × 0.9848)

= 25.63

Therefore, the radius of the circle is 25.63. Please watch the video for more details.