Question

A triangle has side lengths of left parenthesis, r, minus, 10, s, right parenthesis(r−10s) centimeters, left parenthesis, 3, r, plus, 2, t, right parenthesis(3r+2t) centimeters, and left parenthesis, 2, t, minus, 6, s, right parenthesis(2t−6s) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

Answer 1

To find the perimeter of the triangle with the given side lengths, we need to sum the expressions for the lengths of the three sides:

1. The first side is \( r - 10s \) centimeters.
2. The second side is \( 3r + 2t \) centimeters.
3. The third side is \( 2t - 6s \) centimeters.

To calculate the perimeter \( P \), we add these three expressions together:

\[
P = (r - 10s) + (3r + 2t) + (2t - 6s)
\]

Now we will combine like terms. Start by combining the \( r \) terms:

\[
r + 3r = 4r
\]

Next, we combine the \( s \) terms:

\[
-10s - 6s = -16s
\]

Finally, we combine the \( t \) terms:

\[
2t + 2t = 4t
\]

Now we can put it all together:

\[
P = 4r + 4t - 16s
\]

Thus, the expression that represents the perimeter of the triangle, in centimeters, is:

\[
\boxed{4r + 4t - 16s}
\]