If you increase each side length in a triangle by 5%, does this increase the perimeter by
5%? Explore and write up your findings.

### Investigation of a Triangle's Perimeter with a 5% Increase in Side Lengths

#### Problem Setup:
We are asked whether increasing each side length of a triangle by 5% results in a 5% increase in the triangle's perimeter.

#### Definitions and Assumptions:
Let the sides of a triangle be denoted as \(a\), \(b\), and \(c\). The perimeter \(P\) of the triangle is given by the sum of its sides:
\[
P = a + b + c
\]
When we increase each side length by 5%, the new side lengths become:
\[
a' = a \times 1.05, \quad b' = b \times 1.05, \quad c' = c \times 1.05
\]
The new perimeter \(P'\) is the sum of the new side lengths:
\[
P' = a' + b' + c' = 1.05a + 1.05b + 1.05c = 1.05(a + b + c) = 1.05P
\]

#### Findings:
From the above expression, we can see that the new perimeter \(P'\) is exactly 1.05 times the original perimeter \(P\), which means the perimeter increases by 5%.

Thus, **increasing each side of a triangle by 5% results in a 5% increase in its perimeter**.

### General Observation:
This result holds because perimeter is a linear function of the side lengths. When every side length is scaled by the same factor, the perimeter is scaled by the same factor as well.

#### Summary:
- The perimeter of a triangle is proportional to its side lengths.
- A uniform percentage increase in the side lengths results in the same percentage increase in the perimeter.
- Therefore, a 5% increase in each side length leads to a 5% increase in the perimeter.

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Would you like further details or clarification on any part of this explanation? Here are some related questions to explore further:

1. How does the area of the triangle change with a 5% increase in side lengths?
2. Does the same principle apply to other polygons (like quadrilaterals or pentagons)?
3. If only two sides of the triangle increase by 5%, what happens to the perimeter?
4. How does the perimeter change if the sides increase by different percentages?
5. What happens to the perimeter if the sides are decreased by 5% instead of increased?

**Tip:** When dealing with geometric figures, any linear dimension change (like side lengths) will affect other linear quantities (like perimeter) in the same proportional manner.

If you increase each side length in a triangle by 5%, does this increase the perimeter by 5%? Explore and write up your findings.

Investigate whether increasing the side lengths of a triangle by 5% results in a 5% increase in the perimeter. This problem covers core geometry concepts related to perimeter and scaling, including proportional relationships in shapes. Suitable for Grades 7-10.

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