To solve the problem of determining the ratio of a pair of corresponding sides of two similar polygons with areas of 4 square inches and 64 square inches, let's go through the steps:
1. Understanding the Area Ratio:
- First, compute the ratio of the areas of the two polygons. We know that the areas are 4 square inches and 64 square inches. So the ratio of the areas (area of the smaller polygon to the area of the larger polygon) is:
[tex]\[
\text{Ratio of areas} = \frac{4}{64}
\][/tex]
2. Calculate the Ratio of the Areas:
- Simplify the ratio of the areas:
[tex]\[
\frac{4}{64} = \frac{1}{16}
\][/tex]
3. Understanding Side Ratio:
- For similar polygons, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Let's denote the ratio of the corresponding sides by [tex]\( x \)[/tex]. Therefore,
[tex]\[
x^2 = \frac{1}{16}
\][/tex]
4. Solving for the Ratio of Corresponding Sides:
- To find the ratio of the corresponding sides ([tex]\( x \)[/tex]), take the square root of the ratio of the areas:
[tex]\[
x = \sqrt{\frac{1}{16}} = \frac{1}{4}
\][/tex]
Therefore, the ratio of a pair of corresponding sides of the two similar polygons is [tex]\( \frac{1}{4} \)[/tex].
Given the options, the correct answer is:
- The ratio of a pair of corresponding sides is [tex]\( \frac{1}{4} \)[/tex].
