Question
When you have two shapes to compare on a coordinate plane, you can determine the scale factor, knowing that the transformation was a dilation. Generate instructions you would give another student to determine the scale factor.
Answers
GPT-4o mini
To determine the scale factor between two shapes that have been dilated on a coordinate plane, follow these steps:
### Instructions to Determine the Scale Factor
1. **Identify the Original and Dilated Shapes:**
- Label the points of the original shape as \( A \), \( B \), \( C \), etc. and the corresponding points of the dilated shape as \( A' \), \( B' \), \( C' \), etc.
2. **Choose Corresponding Points:**
- Select a pair of corresponding points from the original shape and the dilated shape. For example, you could choose points \( A \) and \( A' \).
3. **Determine the Coordinates:**
- Write down the coordinates of these points. For example, let \( A(x_1, y_1) \) be the coordinate of point \( A \) in the original shape, and \( A'(x_2, y_2) \) be the coordinate of point \( A' \) in the dilated shape.
4. **Calculate the Distance from the Center of Dilation:**
- Identify the center of dilation. Denote this point as \( O(h, k) \).
- Use the distance formula to calculate the distance from the center of dilation \( O \) to point \( A \) as follows:
\[
d_A = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\]
- Similarly, calculate the distance from the center of dilation to the dilated point \( A' \):
\[
d_{A'} = \sqrt{(x_2 - h)^2 + (y_2 - k)^2}
\]
5. **Calculate the Scale Factor:**
- The scale factor \( k \) can be determined by using the following formula:
\[
k = \frac{d_{A'}}{d_A}
\]
- This will give you the scale factor that represents how much the original shape has been enlarged or reduced to get to the dilated shape.
6. **Repeat for Other Points (Optional):**
- To ensure accuracy, you can repeat this process with other pairs of corresponding points (e.g., \( B \) and \( B' \), \( C \) and \( C' \)) and calculate the scale factor for each pair. The value should remain consistent if the transformation is a true dilation.
7. **Conclude:**
- If all calculated scale factors are the same, then you've validated the dilation process, and you can confidently state the scale factor for the transformation.
### Example
Assume:
- Original point \( A(1, 2) \)
- Dilated point \( A'(4, 8) \)
- Center of dilation \( O(0, 0) \)
1. Calculate \( d_A \):
\[
d_A = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
2. Calculate \( d_{A'} \):
\[
d_{A'} = \sqrt{(4 - 0)^2 + (8 - 0)^2} = \sqrt{16 + 64} = \sqrt{80}
\]
3. Calculate the scale factor \( k \):
\[
k = \frac{\sqrt{80}}{\sqrt{5}} = \frac{4\sqrt{5}}{\sqrt{5}} = 4
\]
Thus, the scale factor is 4.
Follow these steps carefully, and you'll be able to determine the scale factor for any pair of dilated shapes!
### Instructions to Determine the Scale Factor
1. **Identify the Original and Dilated Shapes:**
- Label the points of the original shape as \( A \), \( B \), \( C \), etc. and the corresponding points of the dilated shape as \( A' \), \( B' \), \( C' \), etc.
2. **Choose Corresponding Points:**
- Select a pair of corresponding points from the original shape and the dilated shape. For example, you could choose points \( A \) and \( A' \).
3. **Determine the Coordinates:**
- Write down the coordinates of these points. For example, let \( A(x_1, y_1) \) be the coordinate of point \( A \) in the original shape, and \( A'(x_2, y_2) \) be the coordinate of point \( A' \) in the dilated shape.
4. **Calculate the Distance from the Center of Dilation:**
- Identify the center of dilation. Denote this point as \( O(h, k) \).
- Use the distance formula to calculate the distance from the center of dilation \( O \) to point \( A \) as follows:
\[
d_A = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\]
- Similarly, calculate the distance from the center of dilation to the dilated point \( A' \):
\[
d_{A'} = \sqrt{(x_2 - h)^2 + (y_2 - k)^2}
\]
5. **Calculate the Scale Factor:**
- The scale factor \( k \) can be determined by using the following formula:
\[
k = \frac{d_{A'}}{d_A}
\]
- This will give you the scale factor that represents how much the original shape has been enlarged or reduced to get to the dilated shape.
6. **Repeat for Other Points (Optional):**
- To ensure accuracy, you can repeat this process with other pairs of corresponding points (e.g., \( B \) and \( B' \), \( C \) and \( C' \)) and calculate the scale factor for each pair. The value should remain consistent if the transformation is a true dilation.
7. **Conclude:**
- If all calculated scale factors are the same, then you've validated the dilation process, and you can confidently state the scale factor for the transformation.
### Example
Assume:
- Original point \( A(1, 2) \)
- Dilated point \( A'(4, 8) \)
- Center of dilation \( O(0, 0) \)
1. Calculate \( d_A \):
\[
d_A = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
2. Calculate \( d_{A'} \):
\[
d_{A'} = \sqrt{(4 - 0)^2 + (8 - 0)^2} = \sqrt{16 + 64} = \sqrt{80}
\]
3. Calculate the scale factor \( k \):
\[
k = \frac{\sqrt{80}}{\sqrt{5}} = \frac{4\sqrt{5}}{\sqrt{5}} = 4
\]
Thus, the scale factor is 4.
Follow these steps carefully, and you'll be able to determine the scale factor for any pair of dilated shapes!