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The question asks to find the perimeter of a triangle with the side lengths:

- \( 3q + 1 \) centimeters
- \( 5q - 3 \) centimeters
- \( r - 9 \) centimeters

The perimeter of a triangle is the sum of its side lengths. Let's add the three side lengths:

\[
P = (3q + 1) + (5q - 3) + (r - 9)
\]

Now, let's simplify:

\[
P = 3q + 1 + 5q - 3 + r - 9
\]

Group the terms:

\[
P = (3q + 5q) + (1 - 3 - 9) + r
\]

\[
P = 8q + (-11) + r
\]

\[
P = 8q - 11 + r
\]

So, the expression that represents the perimeter is:

\[
\boxed{8q - 11 + r}
\]

The correct answer is the first option: \( 8q - 11 + r \).

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would the formula change if the triangle were a right triangle?
2. Can the perimeter formula be applied to polygons with more than three sides?
3. What happens to the perimeter if the value of \( q \) is doubled?
4. How would you find the perimeter if one side was an unknown variable instead?
5. How does the perimeter formula differ from the area formula for a triangle?

**Tip**: When simplifying expressions, always group like terms to make calculations easier.

A triangle has side lengths of (3q + 1) centimeters, (5q - 3) centimeters, and (r - 9) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

Learn how to calculate the perimeter of a triangle with side lengths (3q + 1), (5q - 3), and (r - 9). This example involves algebraic simplification and understanding of basic geometry. Ideal for students in Grades 7-10.

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