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.