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

To explore whether increasing each side of a triangle by 5% results in a 5% increase in the perimeter, let's break down the problem systematically.

### Step 1: Define the triangle and its perimeter
Let the original side lengths of the triangle be \(a\), \(b\), and \(c\). The perimeter \(P\) of the triangle is the sum of its side lengths:

\[
P = a + b + c
\]

### Step 2: Increase each side length by 5%
When we increase each side length by 5%, the new side lengths become:

\[
\text{New side length of } a = a \times (1 + 0.05) = 1.05a
\]
\[
\text{New side length of } b = b \times (1 + 0.05) = 1.05b
\]
\[
\text{New side length of } c = c \times (1 + 0.05) = 1.05c
\]

### Step 3: Calculate the new perimeter
The new perimeter, \(P'\), is the sum of the increased side lengths:

\[
P' = 1.05a + 1.05b + 1.05c
\]
Factor out the 1.05:

\[
P' = 1.05(a + b + c)
\]

### Step 4: Compare the new and old perimeters
We know that the original perimeter \(P = a + b + c\). Substituting this into the equation for the new perimeter:

\[
P' = 1.05P
\]

### Step 5: Conclusion
From the equation \(P' = 1.05P\), we can see that the new perimeter is 1.05 times the original perimeter. This shows that **the perimeter increases by exactly 5%** when each side length is increased by 5%.

### Findings:
- When each side of a triangle is increased by 5%, the perimeter of the triangle also increases by 5%.
- This result holds true regardless of the shape of the triangle or the relative lengths of the sides, as the perimeter is a linear function of the side lengths.

### Do you want any more details or have questions?

Here are 5 questions to expand on this topic:
1. What happens to the area of a triangle when all sides are increased by 5%?
2. Does this linear relationship between side lengths and perimeter hold for other polygons?
3. How does increasing one side by 5%, while keeping others constant, affect the perimeter?
4. How does this principle apply to geometric figures like circles or squares?
5. What happens if you increase side lengths by different percentages?

#### Tip:
When working with proportional changes in geometric figures, always check whether the property (like area or perimeter) depends linearly or non-linearly on the dimensions.

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

Explore how increasing each side of a triangle by 5% affects the perimeter. This problem investigates proportional changes in side lengths and confirms that the perimeter increases by 5%, as shown through geometric reasoning. Suitable for students in Grades 8-10.

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