Determine whether the dilation passes through the center of dilation. (1 point) Responses A′B′¯¯¯¯¯¯¯¯¯¯ passes through the center of dilation because it is an enlargement of AB¯¯¯¯¯¯¯¯ by a scale factor of 4. line segment cap A prime cap b prime passes through the center of dilation because it is an enlargement of line segment cap A cap b by a scale factor of 4. A′B′¯¯¯¯¯¯¯¯¯¯ does not pass through the center of dilation because it is taken to a line parallel to AB¯¯¯¯¯¯¯¯ . Modifying above upper A prime upper B prime with bar does not pass through the center of dilation because it is taken to a line parallel to line segment cap A cap b . A′B′¯¯¯¯¯¯¯¯¯¯ does not pass through the center of dilation because it is a reduction of AB¯¯¯¯¯¯¯¯ by a scale factor of 14 . line segment cap A prime cap b prime does not pass through the center of dilation because it is a reduction of line segment cap A cap b by a scale factor of Start Fraction 1 over 4 End Fraction . A′B′¯¯¯¯¯¯¯¯¯¯ passes through the center of dilation because it is taken to a line parallel to AB¯¯¯¯¯¯¯¯ .

Answer 1

To determine whether a dilation passes through the center of dilation, we need to understand some fundamental properties of dilations.

1. Definition of Dilation: A dilation is a transformation that produces an image that is the same shape as the original but is a different size. The dilation is centered at a point called the center of dilation, and every point of the shape is moved away from (if the scale factor is greater than 1) or toward (if the scale factor is less than 1) this point by a scale factor.

2. Key Property: Under dilation, if a line segment \( AB \) is transformed into \( A'B' \):

   • If the center of dilation lies on the line segment \( AB \), then \( A'B' \) will go through the center of dilation.
   • If the line segment does not contain the center of dilation, \( A'B' \) will be a line parallel to \( AB \).

Now, let's analyze the responses:

• Enlargement or Reduction: Whether an enlargement (scale factor > 1) or a reduction (scale factor < 1) occurs does not affect whether a line passes through the center of dilation unless we know whether the center is on the segment.

• Parallel Lines: If \( A'B' \) is parallel to \( AB \), then it does not pass through the center of dilation.

Based on this understanding, let's evaluate the provided responses:

1. "A′B′ passes through the center of dilation because it is an enlargement of AB by a scale factor of 4."

   • This statement does not guarantee passing through the center unless we know the center is on the segment.

2. "line segment cap A prime cap b prime passes through the center of dilation because it is an enlargement of line segment cap A cap b by a scale factor of 4."

   • Same reasoning as above.

3. "A′B′ does not pass through the center of dilation because it is taken to a line parallel to AB."

   • This may be true depending on the position of the center of dilation.

4. "Modifying above upper A prime upper B prime with bar does not pass through the center of dilation because it is taken to a line parallel to line segment cap A cap b."

   • Same reasoning applies here.

5. "A′B′ does not pass through the center of dilation because it is a reduction of AB by a scale factor of 14."

   • Again, this does not automatically demonstrate the condition related to the center of dilation.

6. "line segment cap A prime cap b prime does not pass through the center of dilation because it is a reduction of line segment cap A cap b by a scale factor of Start Fraction 1 over 4 End Fraction."

   • Same reasoning applies.

7. "A′B′ passes through the center of dilation because it is taken to a line parallel to AB."

   • This statement is contradictory; if \( A'B' \) is parallel to \( AB \), it cannot pass through the center of dilation.

The most accurate response is likely the one regarding \( A′B′ \) not passing through the center of dilation because it becomes a line parallel to \( AB \). However, without specific details about the position of the center of dilation relative to \( AB \), one cannot definitively conclude whether or not it passes through the center. If the center of dilation is not on segment \( AB \), then \( A'B' \) would indeed be parallel to \( AB \).

Thus, the response that states "A′B′ does not pass through the center of dilation because it is taken to a line parallel to AB." represents an accurate assertion under general circumstances where we conclude that the center of dilation is not on the original segment.