1. Problem: Given a triangle with BC = 12 cm and the area of triangle CMN is 9 square cm, find the length of AM. 2. Formula: The area of a triangle is given by $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$. 3. Explanation: Here, CMN is a smaller triangle inside the larger triangle ABC. Since AM is perpendicular to BC, AM acts as the height for triangle ABC and also for triangle CMN. 4. Calculation: Let AM = h cm. Since BC = 12 cm, and assuming M lies on BC, the base of triangle CMN is some segment along BC. But since AM and MN are equal, and AM is perpendicular, the base CM of triangle CMN is half of BC, i.e., 6 cm. 5. Using the area formula for triangle CMN: $$9 = \frac{1}{2} \times 6 \times h$$ 6. Simplify: $$9 = 3h$$ 7. Solve for h: $$h = \frac{9}{3} = 3$$ 8. Therefore, the length of AM is 3 cm.