To determine the scale factor of the side lengths of two similar octagons based on their areas, we can use the relationship between the areas of similar figures and the scale factor.
Here are the steps:
1. Identify the areas:
The area of the first octagon (let's call it Area1) is 4 m².
The area of the second octagon (let's call it Area2) is 25 m².
2. Understand the relationship between areas and side lengths:
For similar figures, the ratio of their areas is the square of the scale factor of their side lengths.
If the scale factor is represented as [tex]\( k \)[/tex], then the relationship can be written as:
[tex]\[
\left( \frac{\text{Area2}}{\text{Area1}} \right) = k^2
\][/tex]
3. Set up the equation with the known areas:
[tex]\[
\left( \frac{25}{4} \right) = k^2
\][/tex]
4. Solve for the scale factor [tex]\( k \)[/tex]:
[tex]\[
k = \sqrt{\left( \frac{25}{4} \right)}
\][/tex]
5. Calculate the ratio:
[tex]\[
\frac{25}{4} = 6.25
\][/tex]
6. Take the square root of 6.25:
[tex]\[
\sqrt{6.25} = 2.5
\][/tex]
Therefore, the scale factor of their side lengths is 2.5.
