Your friend is mistaken in their claim. When a rectangle undergoes dilation by a certain scale factor, the area does not change by the same factor; instead, it changes by the square of the scale factor. Let's go through a detailed explanation, using an example to illustrate this point.
### Example:
1. Initial Dimensions of the Rectangle:
- Length = 10 units
- Width = 5 units
2. Scale Factor:
- Let's choose a scale factor of 2.
### Step-by-Step Calculation:
#### 1. Initial Area Calculation:
- To find the area of the original rectangle, we use the formula:
[tex]\[
\text{Area} = \text{Length} \times \text{Width}
\][/tex]
Therefore,
[tex]\[
\text{Initial Area} = 10 \times 5 = 50 \text{ square units}
\][/tex]
#### 2. New Dimensions After Dilation:
- When dilating a shape, each dimension (length and width) is multiplied by the scale factor.
- New Length = [tex]\( 10 \times 2 = 20 \)[/tex] units
- New Width = [tex]\( 5 \times 2 = 10 \)[/tex] units
#### 3. New Area Calculation:
- Using the new dimensions, the area of the dilated rectangle is:
[tex]\[
\text{New Area} = \text{New Length} \times \text{New Width} = 20 \times 10 = 200 \text{ square units}
\][/tex]
#### 4. Ratio of New Area to Initial Area:
- To see how the area changes with respect to the initial area, we compute:
[tex]\[
\text{Area Ratio} = \frac{\text{New Area}}{\text{Initial Area}} = \frac{200}{50} = 4
\][/tex]
This ratio is the square of the scale factor (2), since
[tex]\[
2^2 = 4
\][/tex]
### Conclusion:
The ratio of the new area to the initial area is equal to the square of the scale factor. In this case, the scale factor was 2, and the ratio of the new area to the initial area is 4, which is [tex]\( 2^2 \)[/tex].
Therefore, the area of a rectangle changes by the square of the scale factor when it is dilated, not by the scale factor itself. So, the correct understanding is that if you dilate a rectangle by a scale factor [tex]\( k \)[/tex], the area changes by a factor of [tex]\( k^2 \)[/tex].
Hence, your friend’s claim is incorrect.
