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Let's start with the geometry question (27). Here's what it asks:

- **Problem**: Write an expression for the perimeter of the given triangle. Then evaluate the perimeter when \(n = 1, 2, 3, 4, 5\).

### Step 1: Identify triangle sides.
From the diagram, the triangle has sides labeled as:

- **4n**
- **5n**
- **6n**

### Step 2: Expression for the perimeter.
The perimeter \( P \) of a triangle is the sum of all its sides. Therefore, the perimeter expression is:

\[
P = 4n + 5n + 6n
\]

Simplify this:

\[
P = 15n
\]

### Step 3: Evaluate the perimeter for \(n = 1, 2, 3, 4, 5\).

- For \(n = 1\): \( P = 15(1) = 15 \)
- For \(n = 2\): \( P = 15(2) = 30 \)
- For \(n = 3\): \( P = 15(3) = 45 \)
- For \(n = 4\): \( P = 15(4) = 60 \)
- For \(n = 5\): \( P = 15(5) = 75 \)

These are the values for the perimeter.

Let me know if you'd like to go through any other questions or want more details on this one! Here are 5 related questions to explore further:

1. How do you find the perimeter of other geometric shapes, like a rectangle?
2. What is the formula for finding the perimeter of a circle (circumference)?
3. How can you generalize the perimeter of polygons with more sides, like a pentagon or hexagon?
4. What is the relationship between the perimeter and area of a triangle?
5. How does the perimeter of a shape change if you scale its dimensions by a factor?

**Tip**: Always check if you can simplify algebraic expressions, especially in geometry problems, to make calculations easier!

Write an expression for the perimeter of the given triangle. Then evaluate the perimeter when n = 1, 2, 3, 4, and 5.

Learn how to find the perimeter of a triangle with side lengths given as 4n, 5n, and 6n. This step-by-step guide shows how to evaluate the perimeter for different values of n, suitable for students in grades 6-8.

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