To find the perimeter of a triangle when its side lengths are given, you simply add up the expressions for each of the three sides.
Given the expressions for the side lengths:
1. [tex]\(6x + 10y\)[/tex] centimeters
2. [tex]\(8x - 9z\)[/tex] centimeters
3. [tex]\(4z - 7y\)[/tex] centimeters
We will add these expressions together to find the perimeter:
[tex]\[
\text{Perimeter} = (6x + 10y) + (8x - 9z) + (4z - 7y)
\][/tex]
Now, let's combine like terms:
1. Combine the [tex]\(x\)[/tex] terms:
[tex]\[
6x + 8x = 14x
\][/tex]
2. Combine the [tex]\(y\)[/tex] terms:
[tex]\[
10y - 7y = 3y
\][/tex]
3. Combine the [tex]\(z\)[/tex] terms:
[tex]\[
-9z + 4z = -5z
\][/tex]
Putting it all together, the perimeter of the triangle is:
[tex]\[
14x + 3y - 5z
\][/tex]
Therefore, the expression that represents the perimeter, in centimeters, of the triangle is:
[tex]\[
14x + 3y - 5z
\][/tex]
Among the given multiple-choice options, the correct answer is:
[tex]\[
\boxed{14x + 3y - 5z}
\][/tex]
