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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 = 80o, AC = 45. What is the radius of the circle?

find the radius of a circle and a triangle inscribed in this 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 × cos80o
= 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 80o)
= 50.5 / (2 × 0.9848)
= 25.63

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