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?

The perimeter of a triangle is the sum of the lengths of all three sides. In this case, the sides of the triangle are given as:

1. \( (r - 10s) \) cm
2. \( (3r + 2t) \) cm
3. \( (2t - 6s) \) cm

To find the perimeter, add these three expressions together:

\[
\text{Perimeter} = (r - 10s) + (3r + 2t) + (2t - 6s)
\]

Now, let's simplify this expression by combining like terms:

- Combine the \( r \)-terms: \( r + 3r = 4r \)
- Combine the \( s \)-terms: \( -10s - 6s = -16s \)
- Combine the \( t \)-terms: \( 2t + 2t = 4t \)

Thus, the expression for the perimeter becomes:

\[
\text{Perimeter} = 4r - 16s + 4t
\]

So, the expression that represents the perimeter of the triangle is:

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

Would you like a detailed explanation of how I combined these terms, or do you have any other questions?

Here are 5 related questions to extend your understanding:
1. What would happen if one of the sides of the triangle was missing?
2. How would the perimeter change if \( r \), \( s \), or \( t \) increased by 1?
3. Can you find the perimeter if specific values for \( r \), \( s \), and \( t \) are given?
4. How would the formula for the perimeter change if all side lengths were multiplied by a constant factor?
5. Could this be a right triangle based on these side lengths?

**Tip:** Always combine like terms when simplifying algebraic expressions to reduce them to their simplest form.

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

Discover how to find the perimeter of a triangle with side lengths expressed as algebraic terms: (r−10s), (3r+2t), and (2t−6s) centimeters. This problem focuses on simplifying algebraic expressions to calculate the perimeter and is suitable for students in Grades 8-10.

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