Question

the areas of two similar octagons are 4 m² and 9 m². what is the scale factor of their side lengths? ?:

Explanation:

Step1: Recall the relationship between areas and scale factor of similar figures

For similar figures, if the scale factor of side lengths is \(k\), the ratio of their areas is \(k^{2}\). Let the areas be \(A_1 = 4\space m^{2}\) and \(A_2=9\space m^{2}\), and the scale factor of side lengths be \(k=\frac{s_1}{s_2}\) (where \(s_1\) and \(s_2\) are side lengths of the two similar octagons). Then \(\frac{A_1}{A_2}=k^{2}\).

Step2: Solve for \(k\)

Substitute \(A_1 = 4\) and \(A_2 = 9\) into \(\frac{A_1}{A_2}=k^{2}\), we get \(k^{2}=\frac{4}{9}\). Take the square - root of both sides. Since \(k>0\) (scale factor is a positive quantity for lengths), \(k=\sqrt{\frac{4}{9}}=\frac{2}{3}\).

Answer:

\(2:3\)