Final answer:
The perimeter of the large rectangle is three times greater than that of the small rectangle, while the area is nine times greater. These differences occur because the perimeter is proportional to the side lengths, while the area is proportional to the square of the scale factor. If the dimensions are doubled, the perimeter would be doubled, and the area would become four times greater.
Step-by-step explanation:
Understanding the Relationship Between Dimensions, Perimeter, and Area
When we increase the dimensions of a rectangle, its perimeter and area also change. Let's explore how these properties are related.
Perimeter of the Rectangle
If the large rectangle's dimensions are three times the small rectangle's dimensions, then the perimeter of the large rectangle will also be three times the perimeter of the small rectangle. This is because the perimeter is a linear measurement and directly proportional to the length of the sides.
Area of the Rectangle
The area, however, will change by a factor of the square of the scale factor. If the scale factor is three, the area of the large rectangle is nine times the area of the small rectangle. That's because area is a two-dimensional measurement, calculated by multiplying length by width.
Comparison Between Perimeter and Area
The answers to parts (a) and (b) are not the same. The perimeter changes linearly while the area changes quadratically with the scale factor.
Scaling by a Factor of Two
If the dimensions of the large rectangle are twice those of the small rectangle, the perimeter of the large rectangle would be twice that of the small one, and the area would be four times as large, following the rule that the ratio of areas of similar figures is the square of the scale factor.