To find the area of the enlarged figure (the larger rectangle) using the scale factor, we first need to establish the scale factor between the small rectangle and the large rectangle.
Step 1: Determine the scale factor
The small rectangle has:
- Length = 8
- Width = 4
The large rectangle has:
- Length = 25.6
- Width = a (unknown)
To find the scale factor based on the lengths: \[ \text{Scale factor} = \frac{\text{Length of large rectangle}}{\text{Length of small rectangle}} = \frac{25.6}{8} \]
Calculating this gives: \[ \text{Scale factor} = \frac{25.6}{8} = 3.2 \]
Step 2: Relate the width of the large rectangle to the scale factor
Since the rectangles are similar, the width of the large rectangle must also follow the same scale factor. Thus, we can find the width \(a\) of the large rectangle using the scale factor: \[ a = \text{Scale factor} \times \text{Width of small rectangle} = 3.2 \times 4 \]
Calculating this gives: \[ a = 3.2 \times 4 = 12.8 \]
Step 3: Calculate the area of the enlarged (large) rectangle
Now we can find the area of the large rectangle using its length and width:
- Length of large rectangle = 25.6
- Width of large rectangle = 12.8
The area \(A\) of the large rectangle is given by the formula: \[ A = \text{Length} \times \text{Width} \]
Substituting the values, we get: \[ A = 25.6 \times 12.8 \]
Calculating this gives: \[ A = 327.68 \]
Step 4: Summary
- We found that the scale factor between the small and large rectangles is 3.2.
- The width of the large rectangle was found to be 12.8.
- Finally, we calculated that the area of the enlarged rectangle is 327.68 square units.
So, the area of the enlarged figure is \(327.68\) square units.