For the next three (03) items that follow : The centre O of a circle inside a triangle ABC is at a distance of 13 cm from each of the vertices of the triangle. The diameter of the circle is 10 cm and the circle touches only two sides of the triangle, AB and AC. If x is the perimeter, in cm, of the triangle, then which one of the following is correct?
- A63 cm < x < 65 cm
- B65 cm < x < 67 cm
- C67 cm < x < 69 cm
- D69 cm < x < 71 cm
Explanation
O is the circumcenter with circumradius R = 13 (equidistant from all vertices). The circle of radius 5 centred at O touches AB and AC, so the distances from O to these sides are 5, i.e. R·cos C = 5 and R·cos B = 5, giving cos B = cos C = 5/13. Hence b = c = 2R·sin B = 26·(12/13) = 24 and a = 2R·sin A = 26·sin 2B = 26·(120/169) = 3120/169 ≈ 18.46. The perimeter x ≈ 66.46 lies in (65, 67), option B.