Use the image to answer the question. 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 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¯¯¯¯¯¯¯¯ . Modifying above upper A prime upper B prime with bar passes through the center of dilation because it is taken to a line parallel to Modifying above upper A upper B with bar . A′B′¯¯¯¯¯¯¯¯¯¯ does not pass through the center of dilation because it is taken to a line parallel to AB¯¯¯¯¯¯¯¯ .

Answer 1

To determine whether the dilation passes through the center of dilation, we need to consider the properties of dilation in geometry.

1. A dilation transforms a figure into a similar figure, enlarging or reducing its size based on a scale factor relative to a fixed point called the center of dilation.
2. If the original line segment \( AB \) is dilated from a point \( O \) to form the line segment \( A'B' \), the line segment \( A'B' \) will pass through the center of dilation \( O \) if and only if the dilation takes points on line segment \( AB \) and extends them to points on the ray extending out from \( O \).

From the responses provided, we can summarize:

• If \( A'B' \) is an enlargement by a scale factor of 4, it will not necessarily indicate whether \( A'B' \) passes through the center of dilation unless it is specified that it does so from the center.
• If \( A'B' \) represents a reduction of \( AB \), it won't pass through the center if it does not extend in the direction from the center forward, suggesting a reduction.
• If \( A'B' \) is parallel to \( AB \), it indicates that the dilation does not align through the center of dilation.

Thus, based on these principles, you would select one of the responses that correctly identifies that \( A'B' \) either passes through the center of dilation or does not based on the scale factor and nature (enlargement or reduction) given in the question.

If I had to pick from the options you provided, I would say the fourth response seems the most accurate because it refers to line segments being taken to lines parallel, which indicates that they do not intersect at the center of dilation.

"The line segment \( A'B' \) does not pass through the center of dilation because it is taken to a line parallel to \( AB \)."