Problem 21 - Prealgebra

Question

Use the two rectangles. The ratio comparing their sides is 1: 2. Write the ratio that compares their areas. How does the ratio relate to the scale factor of the small rectangle to the large rectangle?

Step-by-Step Solution

Answer

The area ratio is 1:4, corresponding to the square of the side scale factor (1:2).

Step 1: Understand the Side Ratio

We are given that the ratio of the sides of the small rectangle to the large rectangle is 1:2. This means if the small rectangle has a side length of \( a \), then the corresponding side of the large rectangle is \( 2a \).

Step 2: Calculate Area of Small Rectangle

Assume the sides of the small rectangle are \( a \) and \( b \). The area of the small rectangle is given by the formula: \( A_{small} = a \times b \).

Step 3: Calculate Area of Large Rectangle

The sides of the large rectangle are \( 2a \) and \( 2b \) due to the ratio 1:2. The area of the large rectangle is: \( A_{large} = 2a \times 2b = 4ab \).

Step 4: Find Ratio of Areas

To find the ratio of the areas of the small rectangle to the large rectangle, divide their areas: \( \text{Ratio}_{area} = \frac{A_{small}}{A_{large}} = \frac{ab}{4ab} = \frac{1}{4} \).

Step 5: Relate to Scale Factor

The scale factor from the small rectangle to the large rectangle for the sides is 2. The area ratio, \( \frac{1}{4} \), is the square of the side scale factor, \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). This demonstrates that the ratio of the areas is the square of the scale factor of the corresponding sides.

Key Concepts

Rectangle Area Calculation

Calculating the area of a rectangle is a fundamental concept in geometry. The area of a rectangle is simply the amount of space enclosed within the boundary of the rectangle.
It is measured in square units. The formula to find the area is:

• Area = Length × Width

For example, if the sides of a rectangle are "a" and "b", then the area is represented as:

• \( A = a \times b \)

Calculating the area helps us compare different rectangles and solve real-world problems like determining the necessary materials for construction. Knowing how to calculate rectangle area accurately is crucial to solving related geometry exercises.
It's simple once you know the side lengths – just multiply them together!

Scale Factor

A scale factor is a number used to increase or decrease the dimensions of a shape. In geometry, this concept is essential for understanding how figures can be resized proportionally.
A scale factor compares the size of the new figure to the original, expressed as a ratio. For example, if you have a scale factor of 2, it means every dimension of the original shape is doubled.

• Scale Factor = New Size / Original Size

In the context of rectangles, if the scale factor between two rectangles is 2, this implies each side of the second rectangle is twice as long as the corresponding side of the first rectangle.
Understanding scale factors helps in scaling figures for drawings, models and resizing images while maintaining proportions.

Ratio of Sides

The ratio of the sides compares the corresponding lengths of two shapes. It shows the proportional relationship between the dimensions of similar geometric figures. The side ratio is expressed as a pair of numbers, indicating how many times larger or smaller one shape is compared to another.
If the ratio of the sides of two rectangles is given as 1:2, it suggests that every side of the second rectangle is twice the length of each corresponding side in the first rectangle.

• Side Ratio = Side length of one shape / Side length of another shape

Ratios of sides help in understanding the concept of similar figures, where shapes have the same form but different sizes. It’s a key idea in problems involving geometric transformations, scaling, and resizing objects uniformly. Understanding the side ratio is crucial in determining how such transformations impact area and volume.