Question
The K or the multiplier of a dilation
Answers
GPT-4o mini
In the context of geometry, a dilation is a transformation that alters the size of a figure while maintaining its shape. The ratio by which the figure is enlarged or reduced is referred to as the **scale factor** or **dilation factor**, often denoted by the letter \( k \).
### Properties of Dilation:
1. **Scale Factor (k)**:
- If \( k > 1 \), the dilation enlarges the figure.
- If \( 0 < k < 1 \), the dilation reduces the figure.
- If \( k = 1 \), the figure remains the same size.
- If \( k < 0 \), the figure is reflected and scaled.
2. **Dilation Center**: The transformation is typically defined relative to a specific point in space called the center of dilation.
3. **Coordinates**: If a point \( (x, y) \) is dilated by a factor of \( k \) around the origin (0, 0), the new coordinates \( (x', y') \) after dilation can be computed as:
\[
(x', y') = (kx, ky)
\]
If the center of dilation is a different point \( (a, b) \), the formula adjusts to:
\[
(x', y') = (a + k(x - a), b + k(y - b))
\]
### Example:
- If you have a triangle with vertices at \( A(1, 1) \), \( B(2, 0) \), and \( C(0, 2) \) and you dilate it by a factor of \( k = 2 \) around the origin, the new vertices would be:
- \( A'(2, 2) \)
- \( B'(4, 0) \)
- \( C'(0, 4) \)
In summary, the letter \( k \) in the context of dilation refers to the scale factor that determines how much the figure will be enlarged or reduced during the process.
### Properties of Dilation:
1. **Scale Factor (k)**:
- If \( k > 1 \), the dilation enlarges the figure.
- If \( 0 < k < 1 \), the dilation reduces the figure.
- If \( k = 1 \), the figure remains the same size.
- If \( k < 0 \), the figure is reflected and scaled.
2. **Dilation Center**: The transformation is typically defined relative to a specific point in space called the center of dilation.
3. **Coordinates**: If a point \( (x, y) \) is dilated by a factor of \( k \) around the origin (0, 0), the new coordinates \( (x', y') \) after dilation can be computed as:
\[
(x', y') = (kx, ky)
\]
If the center of dilation is a different point \( (a, b) \), the formula adjusts to:
\[
(x', y') = (a + k(x - a), b + k(y - b))
\]
### Example:
- If you have a triangle with vertices at \( A(1, 1) \), \( B(2, 0) \), and \( C(0, 2) \) and you dilate it by a factor of \( k = 2 \) around the origin, the new vertices would be:
- \( A'(2, 2) \)
- \( B'(4, 0) \)
- \( C'(0, 4) \)
In summary, the letter \( k \) in the context of dilation refers to the scale factor that determines how much the figure will be enlarged or reduced during the process.