Question

texes mathematics 4-8 (115)

1. a rectangular soccer field and a rectangular garden are similar in shape. the dimensions of the soccer field are 3.2 times the corresponding dimensions of the garden. which statement is true about the two rectangular shapes?

a. the perimeter of the soccer field is 6.4 times the perimeter of the garden.
b. the perimeter of the soccer field is 10.24 times the perimeter of the garden.
c. the area of the soccer field is 6.4 times the area of the garden.
d. the area of the soccer field is 10.24 times the area of the garden.

Explanation:

🆕 New Concept Discovered: Area and Perimeter of Similar Figures
How scaling dimensions affects perimeter and area.

Step 1: Analyze the scale factor

The problem states that a rectangular soccer field and a rectangular garden are similar. The dimensions (length and width) of the soccer field are \( 3.2 \) times the corresponding dimensions of the garden.

This means the linear scale factor \( k \) from the garden to the soccer field is:

$$ k = 3.2 $$

Step 2: Determine the relationship between perimeters

The perimeter of a 2D shape is a one-dimensional (linear) measurement. When a figure is scaled by a factor of \( k \), its perimeter scales by that same factor \( k \).

$$ \text{Perimeter}_{\text{field}} = k \times \text{Perimeter}_{\text{garden}} $$

$$ \text{Perimeter}_{\text{field}} = 3.2 \times \text{Perimeter}_{\text{garden}} $$

Therefore, the perimeter of the soccer field is \( 3.2 \) times the perimeter of the garden. This rules out options A and B.

Step 3: Determine the relationship between areas

Area is a two-dimensional measurement. When a figure is scaled by a linear factor of \( k \), its area scales by the square of that factor, \( k^2 \).

$$ \text{Area}_{\text{field}} = k^2 \times \text{Area}_{\text{garden}} $$

$$ \text{Area}_{\text{field}} = (3.2)^2 \times \text{Area}_{\text{garden}} $$

Calculate \( (3.2)^2 \):

$$ 3.2 \times 3.2 = 10.24 $$

$$ \text{Area}_{\text{field}} = 10.24 \times \text{Area}_{\text{garden}} $$

Therefore, the area of the soccer field is \( 10.24 \) times the area of the garden. This matches option D.

Answer:

D. The area of the soccer field is 10.24 times the area of the garden.