To determine the scale factor of the side lengths of two similar octagons, given their areas, follow these steps:
1. Identify the areas of the two similar octagons:
- The area of the smaller octagon is [tex]\( 9 \, m^2 \)[/tex].
- The area of the larger octagon is [tex]\( 25 \, m^2 \)[/tex].
2. Understand the relationship between areas and side lengths for similar figures:
- For similar figures, the ratio of their areas is the square of the ratio of their corresponding side lengths. If [tex]\( k \)[/tex] is the scale factor of the side lengths, then [tex]\( k^2 \)[/tex] is the scale factor of the areas.
3. Set up the equation based on the areas:
- Let [tex]\( k \)[/tex] be the scale factor of the side lengths.
- Since the areas are [tex]\( 9 \, m^2 \)[/tex] and [tex]\( 25 \, m^2 \)[/tex], the ratio of the areas is [tex]\( \frac{25}{9} \)[/tex].
4. Establish the equation involving [tex]\( k \)[/tex]:
- [tex]\( k^2 = \frac{25}{9} \)[/tex]
5. Solve for [tex]\( k \)[/tex]:
- To find [tex]\( k \)[/tex], take the square root of both sides of the equation:
[tex]\[
k = \sqrt{\frac{25}{9}}
\][/tex]
6. Calculate the scale factor:
- Evaluate the square root:
[tex]\[
k = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3} \approx 1.6667
\][/tex]
Therefore, the scale factor of the side lengths of the two similar octagons is approximately [tex]\( 1.6667 \)[/tex].
Additionally, the scale factor of the areas ([tex]\( k^2 \)[/tex]) is:
[tex]\[
\frac{25}{9} \approx 2.7778
\][/tex]
