Main Answer:
Possible scale factors for similar polygons are rational numbers. When polygons are similar, their corresponding sides are in proportion, allowing the scale factor to be expressed as a fraction.
Step-by-step explanation:
Similar polygons have corresponding angles that are equal, and their corresponding sides are in proportion. The scale factor, which represents the ratio of corresponding side lengths, can be any rational number. This is because a rational number is the ratio of two integers, and when you scale each side of a polygon by a rational number, the resulting lengths will also be rational.
For example, if the scale factor is 1/2, it means that each side of the smaller polygon is half the length of the corresponding side in the larger polygon. Since the lengths are rational, the scale factor is rational.
It's important to note that irrational scale factors are not possible for similar polygons. This is because irrational numbers, such as the square root of 2, cannot be expressed as a simple ratio of two integers. Therefore, for polygons to be similar, the scale factor must be rational.
In summary, the scale factors for similar polygons are rational numbers, ensuring that the corresponding sides are in proportion with ratios that can be expressed as fractions of integers.