Find the perimeter of triangle ABC.

Okay, let's solve the problem step by step. 1. Understanding the problem: * Triangle ABC is isosceles with AC as the base. * A circle is inscribed in the triangle, touching sides AB, BC, and AC. * BD = 4, where D is the point of tangency on BC. * AN = 9, where N is the point of tangency on AB. * We need to find the perimeter of triangle ABC. 2. Using properties of tangents: * Tangents from a point to a circle have equal lengths. Therefore: * AN = AM = 9 (where M is the point of tangency on AC) * BD = BE = 4 (where E is the point of tangency on BC) 3. Finding sides of the triangle: * Since the triangle is isosceles, AB = BC. Let's denote AB = BC = x. * We can express AB and BC in terms of known segments: * AB = AN + NB = 9 + NB * BC = BD + DC = 4 + DC * Since AB = BC, then 9 + NB = 4 + DC. * Also tangents from point B: NB = BD = 4. (where D is the point of tangency on BC) * Then: AB = AN + NB = 9 + 4 = 13 * AB = BC = 13 * Since AM = 9 and MC = DC, we have AC = AM + MC * DC = BC - BD = 13 - 4 = 9 * Since MC = DC, then MC = 9 * Therefore, AC = AM + MC = 9 + 9 = 18 4. Calculating the perimeter: * The perimeter P of triangle ABC is P = AB + BC + AC * P = 13 + 13 + 18 = 44 Answer: 44