If x is the perimeter, in cm, of the triangle, then which one of the following is correct?
\(65\text{ cm} < x < 67\text{ cm}\)
Since O is 13 cm from every vertex, O is the circumcentre with circumradius R = 13 cm. The circle of radius 5 cm touches AB and AC only, so the perpendicular distances from O to AB and AC both equal 5 cm, i.e. R cos C = R cos B = 5, giving cos B = cos C = 5/13 and hence B = C (triangle is isosceles with AB = AC). So sin B = 12/13, AB = AC = 2R sin B = 24 cm. Also A = 180 deg - 2B, so sin A = 2 sin B cos B = 120/169 and cos A = 1 - 2cos^2B = 119/169. Side BC = 2R sin A = 2(13)(120/169) = 3120/169 \(\approx 18.46\text{ cm}\). So perimeter \(x = AB+AC+BC = 24+24+18.46 \approx 66.46\text{ cm}\), which lies between 65 cm and 67 cm.
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