A triangle has side lengths of left parenthesis, 3, point, 8, k, minus, 8, point, 4, m, right parenthesis(3.8k−8.4m) centimeters, left parenthesis, 4, point, 4, k, minus, 4, point, 7, n, right parenthesis(4.4k−4.7n) centimeters, and left parenthesis, 1, point, 7, n, plus, 1, point, 2, m, right parenthesis(1.7n+1.2m) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

To find the perimeter of the triangle, we need to add up the expressions representing each side length.

The side lengths are:
1. \( 3.8k - 8.4m \) cm
2. \( 4.4k - 4.7n \) cm
3. \( 1.7n + 1.2m \) cm

The perimeter, \( P \), of the triangle is the sum of these three side lengths:

\[
P = (3.8k - 8.4m) + (4.4k - 4.7n) + (1.7n + 1.2m)
\]

Now, let's combine like terms:

1. **Combine the \( k \)-terms**: \( 3.8k + 4.4k = 8.2k \)
2. **Combine the \( m \)-terms**: \( -8.4m + 1.2m = -7.2m \)
3. **Combine the \( n \)-terms**: \( -4.7n + 1.7n = -3n \)

Now substitute these into the expression for \( P \):

\[
P = 8.2k - 7.2m - 3n
\]

So, the expression that represents the perimeter of the triangle is:

\[
\boxed{8.2k - 7.2m - 3n}
\]

Would you like further explanation on simplifying expressions, or do you have any questions?

### Additional Questions to Explore
1. How do you determine the perimeter of polygons with expressions for sides?
2. What are like terms, and why can they be combined?
3. How can you check if a triangle is valid using side lengths?
4. How does the perimeter formula change if the triangle has a constant term for each side length?
5. How do you approach algebraic simplification with variables in multiple terms?

**Tip:** Always check that expressions are fully simplified by combining all like terms.

A triangle has side lengths of (3.8k - 8.4m) centimeters, (4.4k - 4.7n) centimeters, and (1.7n + 1.2m) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

Learn how to find the perimeter of a triangle with variable side lengths expressed in terms of k, m, and n. This problem demonstrates how to simplify algebraic expressions to obtain the perimeter. Suitable for Grades 8-10 students.

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