Use the image to answer the question.

Question

A coordinate plane's x-axis ranges from 0 to 13 and its y-axis ranges from negative 5 to 0, both by 1-unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted forming 2 line segments, 1 solid and 1 dotted.

Determine whether the dilation passes through the center of dilation.

(1 point)
Responses

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¯¯¯¯¯¯¯¯.
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′¯¯¯¯¯¯¯¯¯¯ passes through the center of dilation because it is an enlargement of AB¯¯¯¯¯¯¯¯ by a scale factor of 4.

Answer 1

To determine whether the dilation passes through the center of dilation, we need to analyze the relationship between the original line segment \( AB \) and the dilated line segment \( A'B' \).

If \( A'B' \) is a dilation of \( AB \), the following statements can help:

• If the line segments \( A'B' \) and \( AB \) are parallel, it implies that the dilation is centered at a point which does not lie on \( AB \), and thus \( A'B' \) does not pass through the center of dilation.
• If \( A'B' \) is a scaled version of \( AB \) (either enlarged or reduced) and does not intersect \( AB \) at the center of dilation, then it would also indicate that it does not pass through that center.

Given that, the answer to whether the dilation passes through the center of dilation will depend mostly on the relationship of the positions of the points before and after dilation.

From the choices you've presented, the most fitting response, assuming that \( A'B' \) is a reduction of \( AB \) and does not pass through the center of dilation, is:

A′B′ does not pass through the center of dilation because it is a reduction of AB by a scale factor of \( \frac{1}{4} \).

This conclusion is drawn from the interpretation that \( A'B' \) is slightly smaller than \( AB \) and is parallel to \( AB \), hence confirming that it does not intersect at the dilation center.