1. **Problem statement:** Given triangle ABC with D midpoint of BC, E midpoint of AC, and M the intersection of AD and BE. Given lengths: $AD=6$ cm, $AB=9$ cm, $BE=9$ cm. Find the perimeter of triangle $MDE$. 2. **Key properties:** - D and E are midpoints, so $BD=DC$ and $AE=EC$. - M is the intersection of medians AD and BE. - The medians intersect at centroid M, which divides each median in ratio 2:1 from vertex. 3. **Find lengths of segments in triangle $MDE$:** - Since $D$ and $E$ are midpoints, $DE$ is a mid-segment of triangle $ABC$. - $DE$ is parallel to $AB$ and $DE = \frac{1}{2} AB = \frac{1}{2} \times 9 = 4.5$ cm. 4. **Find $MD$ and $ME$:** - M divides median $AD$ in ratio 2:1, so $AM = \frac{2}{3} AD = \frac{2}{3} \times 6 = 4$ cm and $MD = \frac{1}{3} AD = 2$ cm. - Similarly, M lies on median $BE$. Since $BE=9$ cm, $BM = \frac{2}{3} BE = 6$ cm and $ME = \frac{1}{3} BE = 3$ cm. 5. **Calculate perimeter of $\triangle MDE$:** $$\text{Perimeter} = MD + DE + ME = 2 + 4.5 + 3 = 9.5 \text{ cm}$$ **Final answer:** The perimeter of triangle $MDE$ is $9.5$ cm.