Question
Does the dilation pass through the center of dilation? Choose the best answer below.
Coordinate A (4,-2)
Coordinate B (12, -4)
Coordinate A' (1,-0.5)
Coordinate B' (3,-2)
A. line A' B' passes through the center of dilation because it is taken a line parallel to line AB
B. line A'B' passes through the center of dilation because it is an enlargement of line AB by a scale factor of 4
C. line A'B' does not pass through the center of dilation because it is a reduction of line AB a scale factor of 1/4
D. line A'B' does not pass through the center of dilation because it is taken a line parallel to line AB
Coordinate A (4,-2)
Coordinate B (12, -4)
Coordinate A' (1,-0.5)
Coordinate B' (3,-2)
A. line A' B' passes through the center of dilation because it is taken a line parallel to line AB
B. line A'B' passes through the center of dilation because it is an enlargement of line AB by a scale factor of 4
C. line A'B' does not pass through the center of dilation because it is a reduction of line AB a scale factor of 1/4
D. line A'B' does not pass through the center of dilation because it is taken a line parallel to line AB
Answers
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Answered by
GPT-4o mini
To determine whether line A'B' passes through the center of dilation based on the coordinates provided, let's analyze the situation:
1. **Coordinates**:
- A (4, -2)
- B (12, -4)
- A' (1, -0.5)
- B' (3, -2)
2. **Finding the Center of Dilation**:
The center of dilation can be found by determining the scale factor and how the original points map to the new points.
3. **Calculating the Scale Factor**:
- The dilation from A to A':
- From A (4, -2) to A' (1, -0.5) appears to be a reduction.
- The scale factor \(k\) can be calculated using the ratio of the distances from the center of dilation.
- The same applies to B to B'.
4. **Examining Choices**:
- Option A states that line A'B' is parallel to line AB, which doesn’t provide a conclusion about the center of dilation.
- Option B says it’s an enlargement, which is incorrect since we can see that A' is closer to the origin than A; this represents a reduction, not an enlargement.
- Option C mentions that line A'B' does not pass through the center of dilation because it is a reduction by a scale factor of 1/4. This is true as reductions do not pass through the center in this case.
- Option D states that it does not pass through the center because it is parallel. Parallel lines do not definitively indicate the presence or absence of a center of dilation.
Upon review, Option C is the most appropriate choice based on the nature of the dilation indicated by the coordinates.
**Correct Answer: C. line A'B' does not pass through the center of dilation because it is a reduction of line AB by a scale factor of 1/4.**
1. **Coordinates**:
- A (4, -2)
- B (12, -4)
- A' (1, -0.5)
- B' (3, -2)
2. **Finding the Center of Dilation**:
The center of dilation can be found by determining the scale factor and how the original points map to the new points.
3. **Calculating the Scale Factor**:
- The dilation from A to A':
- From A (4, -2) to A' (1, -0.5) appears to be a reduction.
- The scale factor \(k\) can be calculated using the ratio of the distances from the center of dilation.
- The same applies to B to B'.
4. **Examining Choices**:
- Option A states that line A'B' is parallel to line AB, which doesn’t provide a conclusion about the center of dilation.
- Option B says it’s an enlargement, which is incorrect since we can see that A' is closer to the origin than A; this represents a reduction, not an enlargement.
- Option C mentions that line A'B' does not pass through the center of dilation because it is a reduction by a scale factor of 1/4. This is true as reductions do not pass through the center in this case.
- Option D states that it does not pass through the center because it is parallel. Parallel lines do not definitively indicate the presence or absence of a center of dilation.
Upon review, Option C is the most appropriate choice based on the nature of the dilation indicated by the coordinates.
**Correct Answer: C. line A'B' does not pass through the center of dilation because it is a reduction of line AB by a scale factor of 1/4.**
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