Questions: Find the scale factor and ratio of perimeters for a pair of similar octagons with areas 36 ft² and 49 ft². The scale factor is (Simplify your answer.)

Solution Steps

For similar figures, the ratio of their areas is equal to the square of the scale factor. If the areas of two similar octagons are \( A_1 = 36 \, \text{ft}^2 \) and \( A_2 = 49 \, \text{ft}^2 \), then the scale factor \( k \) satisfies: \[ \frac{A_2}{A_1} = k^2 \]

Substitute the given areas into the equation: \[ \frac{49}{36} = k^2 \] Take the square root of both sides to solve for \( k \): \[ k = \sqrt{\frac{49}{36}} = \frac{7}{6} \]

For similar figures, the ratio of their perimeters is equal to the scale factor. Therefore, the ratio of perimeters is: \[ \frac{P_2}{P_1} = k = \frac{7}{6} \]

Final Answer