To find the scale factor of the side lengths of two similar octagons given their areas, we can use the relationship between the areas and the scale factors of similar figures. Here are the steps:
1. Identify the Areas:
The areas of the two similar octagons are [tex]\( 9 \, \text{m}^2 \)[/tex] and [tex]\( 25 \, \text{m}^2 \)[/tex].
2. Understand the Relationship:
For similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. If the ratio of the areas is [tex]\( \frac{\text{Area}_2}{\text{Area}_1} \)[/tex], then the square of the scale factor (ratio of side lengths) [tex]\( k \)[/tex] can be written as:
[tex]\[
k^2 = \frac{\text{Area}_2}{\text{Area}_1}
\][/tex]
3. Compute the Ratio of the Areas:
Substitute the given areas into this relationship:
[tex]\[
k^2 = \frac{25}{9}
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
To find the scale factor [tex]\( k \)[/tex], take the square root of both sides of the equation:
[tex]\[
k = \sqrt{\frac{25}{9}}
\][/tex]
[tex]\[
k = \frac{\sqrt{25}}{\sqrt{9}}
\][/tex]
[tex]\[
k = \frac{5}{3}
\][/tex]
[tex]\[
k \approx 1.6666666666666667
\][/tex]
Thus, the scale factor of the side lengths of the two similar octagons is approximately [tex]\( 1.6666666666666667 \)[/tex].
