Question
a dilation With a scale factor less than one which reduces the size of the original figure
Answers
GPT-4o mini
A dilation with a scale factor less than one is a transformation that reduces the size of a figure. When performing a dilation, every point of the original figure is moved closer to a fixed point known as the center of dilation. The scale factor determines how much the figure is reduced or enlarged.
Here’s what happens in this case:
1. **Center of Dilation**: This is the point from which all other points are scaled. It can be any point in the plane, including a point inside, outside, or on the original figure.
2. **Scale Factor**: A scale factor \( k \) of less than 1 (for example, \( k = 0.5 \)) means that each point of the figure will be moved closer to the center of dilation. If the original distance from a point to the center of dilation is \( d \), after the dilation, the distance will be \( kd \).
3. **Result**: The result of the dilation is a smaller figure that maintains the same shape as the original figure but is proportionately smaller. The angles remain the same, and the ratios of corresponding sides are preserved.
**Example**: If you have a triangle with vertices at \( A(2, 3) \), \( B(4, 5) \), and \( C(3, 1) \), and you perform a dilation with center at the origin \( O(0, 0) \) and scale factor \( k = 0.5 \), the new vertices after dilation would be:
- \( A'(1, 1.5) \) (calculated as \( (0.5 \times 2, 0.5 \times 3) \))
- \( B'(2, 2.5) \) (calculated as \( (0.5 \times 4, 0.5 \times 5) \))
- \( C'(1.5, 0.5) \) (calculated as \( (0.5 \times 3, 0.5 \times 1) \))
The new triangle \( A'B'C' \) is similar to triangle \( ABC \) but reduced in size by half.
Here’s what happens in this case:
1. **Center of Dilation**: This is the point from which all other points are scaled. It can be any point in the plane, including a point inside, outside, or on the original figure.
2. **Scale Factor**: A scale factor \( k \) of less than 1 (for example, \( k = 0.5 \)) means that each point of the figure will be moved closer to the center of dilation. If the original distance from a point to the center of dilation is \( d \), after the dilation, the distance will be \( kd \).
3. **Result**: The result of the dilation is a smaller figure that maintains the same shape as the original figure but is proportionately smaller. The angles remain the same, and the ratios of corresponding sides are preserved.
**Example**: If you have a triangle with vertices at \( A(2, 3) \), \( B(4, 5) \), and \( C(3, 1) \), and you perform a dilation with center at the origin \( O(0, 0) \) and scale factor \( k = 0.5 \), the new vertices after dilation would be:
- \( A'(1, 1.5) \) (calculated as \( (0.5 \times 2, 0.5 \times 3) \))
- \( B'(2, 2.5) \) (calculated as \( (0.5 \times 4, 0.5 \times 5) \))
- \( C'(1.5, 0.5) \) (calculated as \( (0.5 \times 3, 0.5 \times 1) \))
The new triangle \( A'B'C' \) is similar to triangle \( ABC \) but reduced in size by half.