To determine the perimeter of the triangle, we will sum the expressions representing the side lengths of the triangle.
1. The first side length is [tex]\( q + r \)[/tex] cm.
2. The second side length is [tex]\( 5q - 10s \)[/tex] cm.
3. The third side length is [tex]\( 5s - 7r \)[/tex] cm.
To find the perimeter, we add these three expressions together:
[tex]\[ \text{Perimeter} = (q + r) + (5q - 10s) + (5s - 7r) \][/tex]
Let's combine like terms from the sum:
- Combine the [tex]\( q \)[/tex] terms: [tex]\( q + 5q \)[/tex] gives [tex]\( 6q \)[/tex].
- Combine the [tex]\( r \)[/tex] terms: [tex]\( r - 7r \)[/tex] gives [tex]\( -6r \)[/tex].
- Combine the [tex]\( s \)[/tex] terms: [tex]\( -10s + 5s \)[/tex] gives [tex]\( -5s \)[/tex].
So, the combined expression is:
[tex]\[ 6q - 6r - 5s \][/tex]
Thus, the perimeter of the triangle, in centimeters, is represented by the expression:
[tex]\[ 6q - 6r - 5s \][/tex]
Among the given multiple-choice options, the correct expression that matches our result is:
[tex]\[ 6q - 6r - 5s \][/tex]
Thus, the answer is:
[tex]\[ \boxed{6q - 6r - 5s} \][/tex]
