1. The problem asks for the perimeter of a triangle with side lengths given as expressions: $3k + 5m$, $4n - k$, and $9n + m$ centimeters. 2. The perimeter of a triangle is the sum of the lengths of all its sides. 3. Write the expression for the perimeter by adding the three side lengths: $$\text{Perimeter} = (3k + 5m) + (4n - k) + (9n + m)$$ 4. Combine like terms: - Combine $k$ terms: $3k - k = 2k$ - Combine $m$ terms: $5m + m = 6m$ - Combine $n$ terms: $4n + 9n = 13n$ 5. So the perimeter expression simplifies to: $$2k + 6m + 13n$$ 6. Compare this with the answer choices: - $10mn + 8km + 5kn$ (incorrect, terms are products) - $13n + 6m - 4k$ (incorrect, $k$ term sign and coefficient differ) - $13kn + 10mn$ (incorrect, terms are products) - $9m + 4k + 10n$ (incorrect, coefficients differ) None of the options exactly match $2k + 6m + 13n$, so the correct perimeter expression is $2k + 6m + 13n$ which is not listed. Therefore, the perimeter expression is: $$\boxed{2k + 6m + 13n}$$