Answer :
To determine the effect of a dilation on quadrilateral ABCD with a scale factor of \( \frac{1}{2} \) centered at the point \( (2, 2) \), let's carefully consider the concept of dilation:
1. Definition of Dilation:
- Dilation by a scale factor means resizing the shape, keeping the proportions the same but changing its size.
- The center of dilation remains fixed, and every point on the figure moves along the line that connects the point to the center of dilation, either closer to or farther from the center by a proportional amount based on the scale factor.
2. Scale Factor:
- A scale factor of \( \frac{1}{2} \) means that each point on quadrilateral ABCD will be moved to half the distance from the center \( (2, 2) \) compared to its original distance.
3. Impact on Coordinates:
- To understand the impact on a single point, let's consider an original point \( (x, y) \) on quadrilateral ABCD.
- The formula to find the new position \( (x', y') \) after dilation is:
[tex]\[ x' = c_x + (x - c_x) \cdot s \][/tex]
[tex]\[ y' = c_y + (y - c_y) \cdot s \][/tex]
where \( c_x \) and \( c_y \) are the coordinates of the center of dilation, \( x \) and \( y \) are the coordinates of the original point, and \( s \) is the scale factor.
4. Example Calculation:
- Let's apply it to a point \( (x, y) \). Consider point A with original coordinates \( (4, 4) \).
- The center of dilation is \( (2, 2) \) and the scale factor \( s \) is \( \frac{1}{2} \).
- Applying the formula:
[tex]\[ x' = 2 + (4 - 2) \cdot \frac{1}{2} = 2 + 2 \cdot \frac{1}{2} = 2 + 1 = 3 \][/tex]
[tex]\[ y' = 2 + (4 - 2) \cdot \frac{1}{2} = 2 + 2 \cdot \frac{1}{2} = 2 + 1 = 3 \][/tex]
- So, the new coordinates of point A after dilation are \( (3, 3) \).
Thus, the correct statement about the dilation would be that each point on quadrilateral ABCD is moved such that their coordinates become closer to the center (2, 2) by the ratio defined by the scale factor of \( \frac{1}{2} \).
Finally, since [tex]\( (3, 3) \)[/tex] is an example of a point transformed under these conditions, it supports the conclusion that the dilation by [tex]\( \frac{1}{2} \)[/tex] changes each of the corner points of quadrilateral ABCD as described.
1. Definition of Dilation:
- Dilation by a scale factor means resizing the shape, keeping the proportions the same but changing its size.
- The center of dilation remains fixed, and every point on the figure moves along the line that connects the point to the center of dilation, either closer to or farther from the center by a proportional amount based on the scale factor.
2. Scale Factor:
- A scale factor of \( \frac{1}{2} \) means that each point on quadrilateral ABCD will be moved to half the distance from the center \( (2, 2) \) compared to its original distance.
3. Impact on Coordinates:
- To understand the impact on a single point, let's consider an original point \( (x, y) \) on quadrilateral ABCD.
- The formula to find the new position \( (x', y') \) after dilation is:
[tex]\[ x' = c_x + (x - c_x) \cdot s \][/tex]
[tex]\[ y' = c_y + (y - c_y) \cdot s \][/tex]
where \( c_x \) and \( c_y \) are the coordinates of the center of dilation, \( x \) and \( y \) are the coordinates of the original point, and \( s \) is the scale factor.
4. Example Calculation:
- Let's apply it to a point \( (x, y) \). Consider point A with original coordinates \( (4, 4) \).
- The center of dilation is \( (2, 2) \) and the scale factor \( s \) is \( \frac{1}{2} \).
- Applying the formula:
[tex]\[ x' = 2 + (4 - 2) \cdot \frac{1}{2} = 2 + 2 \cdot \frac{1}{2} = 2 + 1 = 3 \][/tex]
[tex]\[ y' = 2 + (4 - 2) \cdot \frac{1}{2} = 2 + 2 \cdot \frac{1}{2} = 2 + 1 = 3 \][/tex]
- So, the new coordinates of point A after dilation are \( (3, 3) \).
Thus, the correct statement about the dilation would be that each point on quadrilateral ABCD is moved such that their coordinates become closer to the center (2, 2) by the ratio defined by the scale factor of \( \frac{1}{2} \).
Finally, since [tex]\( (3, 3) \)[/tex] is an example of a point transformed under these conditions, it supports the conclusion that the dilation by [tex]\( \frac{1}{2} \)[/tex] changes each of the corner points of quadrilateral ABCD as described.
