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Write a simplified polynomial expression for the perimeter of the figure

To find the perimeter of the figure, we need to add up the lengths of all the outer sides. Let's identify and list each side using the labels provided:

1. **Top side**: \( a - 3b \)
2. **Right vertical side (upper)**: \( c \)
3. **Right horizontal side (middle)**: \( 3b \)
4. **Right vertical side (lower)**: \( b + c \)
5. **Bottom side**: \( a - c \)
6. **Left vertical side (lower)**: \( c \)
7. **Left horizontal side (middle)**: \( 2b + 2c \)

Now, let's add all these side lengths to get the perimeter:
\[
\text{Perimeter} = (a - 3b) + c + 3b + (b + c) + (a - c) + c + (2b + 2c)
\]

Next, we'll combine like terms:
- **Terms with \(a\)**: \(a + a = 2a\)
- **Terms with \(b\)**: \(-3b + 3b + b + 2b = 3b\)
- **Terms with \(c\)**: \(c + c - c + c + 2c = 3c\)

So, the simplified expression for the perimeter is:
\[
2a + 3b + 3c
\]

Would you like more details, or do you have any questions about this solution?

Here are five related questions to consider:
1. What would be the perimeter if \( a = 10 \), \( b = 5 \), and \( c = 2 \)?
2. How would the perimeter change if one of the side lengths was doubled?
3. Can you derive a formula for the area of this figure?
4. How would you express the perimeter if the figure was a regular polygon?
5. What happens to the perimeter if all variables \( a \), \( b \), and \( c \) are equal?

**Tip**: Always double-check the figure to ensure you have included all the sides in your perimeter calculation.

Write a simplified polynomial expression for the perimeter of the figure.

Learn how to write a simplified polynomial expression for the perimeter of a composite shape. This problem covers algebraic perimeter calculation and polynomial simplification, suitable for Grades 7-10 students.

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