To determine the expression that represents the perimeter of the triangle, we need to add together the lengths of its three sides. The given side lengths are:
1. [tex]\( q + r \)[/tex] centimeters
2. [tex]\( 5q - 10s \)[/tex] centimeters
3. [tex]\( 5s - 7r \)[/tex] centimeters
Next, we need to find the sum of these three expressions:
[tex]\[
(q + r) + (5q - 10s) + (5s - 7r)
\][/tex]
Let's combine like terms step-by-step:
1. Combine all the terms involving [tex]\( q \)[/tex]:
[tex]\[
q + 5q = 6q
\][/tex]
2. Combine all the terms involving [tex]\( r \)[/tex]:
[tex]\[
r - 7r = -6r
\][/tex]
3. Combine all the terms involving [tex]\( s \)[/tex]:
[tex]\[
-10s + 5s = -5s
\][/tex]
Putting it all together, we get the final simplified expression for the perimeter:
[tex]\[
6q - 6r - 5s
\][/tex]
Thus, the expression that represents the perimeter of the triangle is:
[tex]\[
6q - 6r - 5s
\][/tex]
Therefore, among the provided answer choices, the correct one is:
[tex]\[
\boxed{6 q - 6 r - 5 s}
\][/tex]
