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If 100 regular triangles with side lengths of 1 unit are lined up in a row, sharing one side with the adjacent triangle, what will the perimeter of the shape be?
Consider the following questions:
What will the perimeter of the shape be?
What shape do the 100 triangles create?
Can you create a rule for finding the perimeter of a shape created by any number of triangles?

To calculate the perimeter of 100 regular triangles lined up in a row, we need to consider the properties of the arrangement:

### **Understanding the Problem:**
- Each triangle has three equal sides (equilateral triangles).
- When triangles are placed next to each other, they share one side with their neighbor.
- The key is to figure out how the perimeter is affected by the number of triangles and their shared sides.

### **Step-by-Step Process:**

1. **Single Triangle:**
   A single equilateral triangle has a perimeter of 3 times the side length. If the side length is 1 unit, the perimeter for one triangle is:
   \[
   P = 3 \times 1 = 3 \, \text{units}.
   \]

2. **Two Triangles:**
   When two triangles are placed next to each other sharing one side, they will lose one edge on the perimeter because they share a side. So, instead of the two individual perimeters (which would be 6 units), we lose 1 unit for the shared edge:
   \[
   P = 6 - 1 = 5 \, \text{units}.
   \]

3. **General Rule:**
   Each additional triangle adds two new sides to the perimeter (the left and right side), but one side is shared with the neighboring triangle. Therefore, the perimeter for \( n \) triangles is calculated as:
   \[
   P = 2n + 1.
   \]
   This formula works because the first and last triangles each have two exposed sides, and all other triangles in the middle only add 1 side to the perimeter.

### **Perimeter for 100 Triangles:**

Using the rule \( P = 2n + 1 \), for 100 triangles:
\[
P = 2 \times 100 + 1 = 201 \, \text{units}.
\]

### **Shape of the 100 Triangles:**
The shape created is a zig-zag pattern of triangles, resembling a sawtooth-like figure when viewed from the side. This means that the triangles are aligned in a single line, and the overall figure is elongated horizontally.

### **General Rule for Any Number of Triangles:**
For any number \( n \) of regular triangles, the perimeter can be found using the formula:
\[
P = 2n + 1.
\]

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Do you have any further questions or would you like more details on any part of the solution?

### Related Questions:
1. What would the perimeter be if there were 50 triangles?
2. How does the shape change if the triangles are arranged differently (e.g., stacked)?
3. Can this rule be extended to polygons other than triangles?
4. How does the perimeter change if the side lengths of the triangles are doubled?
5. What happens to the perimeter if the triangles are not regular but isosceles?

### Tip:
Whenever dealing with repeating shapes or patterns, look for the number of shared sides or edges to reduce calculations for perimeter or area.

If 100 regular triangles with side lengths of 1 unit are lined up in a row, sharing one side with the adjacent triangle, what will the perimeter of the shape be?

This page explains how to calculate the perimeter of a shape created by 100 regular triangles aligned in a row. Learn how the perimeter is reduced due to shared sides and find a general formula for calculating the perimeter for any number of triangles. Suitable for students in Grades 5-7.

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