1. **State the problem:** We need to find the perimeter of triangle ABC with an inscribed circle tangent to all three sides. 2. **Given:** - Side AC is split into two segments by the tangency point: 3 cm and 9 cm. - Side BC has a tangency segment of 3 cm from B to the tangency point. - Side AB has a tangency point, but its lengths are unknown. 3. **Key property:** For a triangle with an inscribed circle, the tangents from each vertex to the points of tangency are equal in length. 4. **Assign variables:** - Let the tangents from vertex A be $x$ and $3$ cm (given). - Let the tangents from vertex B be $3$ cm (given) and $y$. - Let the tangents from vertex C be $9$ cm (given) and $y$. Since tangents from the same vertex are equal: - From A: $x = 3$ - From B: $3 = y$ - From C: $9 = y$ But $y$ cannot be both 3 and 9, so we must reassign carefully. 5. **Correct assignment:** - Let the tangents from A be $a$ and $3$ (given segment on AC). - From B: $3$ (given on BC) and $b$ (unknown on AB). - From C: $9$ (given on AC) and $b$ (unknown on BC). Since tangents from the same vertex are equal: - At A: tangents are $a$ and $3$ so $a=3$ - At B: tangents are $3$ and $b$ so $b=3$ - At C: tangents are $9$ and $b$ so $b=9$ Conflict again, so let's label the tangents properly: Let the points of tangency be $D$ on $AC$, $E$ on $AB$, and $F$ on $BC$. - Tangents from A: $AD = AE = x$ - Tangents from B: $BE = BF = y$ - Tangents from C: $CF = CD = z$ Given: - $AD = 3$ cm (from A to tangency on AC) - $DC = 9$ cm (from tangency to C on AC) - $BF = 3$ cm (from B to tangency on BC) So: - $x = AD = 3$ - $z = DC = 9$ - $y = BF = 3$ 6. **Calculate sides:** - $AB = AE + EB = x + y = 3 + 3 = 6$ cm - $BC = BF + FC = y + z = 3 + 9 = 12$ cm - $AC = AD + DC = 3 + 9 = 12$ cm 7. **Perimeter:** $$P = AB + BC + AC = 6 + 12 + 12 = 30$$ cm **Final answer:** The perimeter of the triangle is **30 cm**.