Which statement proves that the dilation of ΔXYZ does not pass through the center of dilation? (1 point) Responses XY¯¯¯¯¯¯¯¯ and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯ both have an undefined slope with a center of dilation at (0,0) . This proves that the lines are parallel and that the dilation does not pass through the center of dilation. Modifying above upper X upper Y with bar and Modifying above upper X prime upper Y prime with bar both have an undefined slope with a center of dilation at left parenthesis 0 comma 0 right parenthesis . This proves that the lines are parallel and that the dilation does not pass through the center of dilation. XY¯¯¯¯¯¯¯¯ has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯ has a length of 1, which proves that ΔXYZ is a dilation of ΔX′Y′Z′ by a scale factor of 3. Modifying above upper X upper Y with bar has a length of 3 and Modifying above upper X prime upper Y prime with bar has a length of 1, which proves that triangle upper X upper Y upper Z is a dilation of triangle upper X prime upper Y prime upper Z prime by a scale factor of 3. XY¯¯¯¯¯¯¯¯ has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯ has a length of 1, which proves that ΔX′Y′Z′ is a dilation of ΔXYZ by a scale factor of 13 . Modifying above upper X upper Y with bar has a length of 3 and Modifying above upper X prime upper Y prime with bar has a length of 1, which proves that triangle upper X prime upper Y prime upper Z prime is a dilation of triangle upper X upper Y upper Z by a scale factor of Start Fraction 1 over 3 End Fraction . X′(−3⋅13,−3⋅13)=X′(−1,−1) , Y′(−3⋅13,−6⋅13)=Y′(−1,−2) , Z′(−6⋅13,−3⋅13)=Z′(−2,−1) . upper X prime left parenthesis negative 3 times Start Fraction 1 over 3 End Fraction comma negative 3 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper X prime left parenthesis negative 1 comma negative 1 right parenthesis , upper Y prime left parenthesis negative 3 times Start Fraction 1 over 3 End Fraction comma negative 6 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper Y prime left parenthesis negative 1 comma negative 2 right parenthesis , upper Z prime left parenthesis negative 6 times Start Fraction 1 over 3 End Fraction comma negative 3 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper Z prime left parenthesis negative 2 comma negative 1 right parenthesis .

Answer 1

To prove that the dilation of ΔXYZ does not pass through the center of dilation, we look for evidence of the relationship between the original triangle and its dilated image.

The relevant statements provided can be analyzed as follows:

1. The first two responses state that XY and X′Y′ both have an undefined slope. This indicates that these two lines are vertical, which confirms that they are parallel. This can imply that the dilation does not pass through the center of dilation because if it did, the segments connecting corresponding points (like XY and X′Y′) would not maintain the same orientation after dilation.

2. The next two statements assert that the lengths of XY and X′Y′ confirm a specific scale factor (3). While this provides information about the scale factor of dilation, it does not directly prove that the dilation does not pass through the center.

3. The last part contains equations for points after scaling. However, this portion primarily shows the coordinates of the dilated points rather than addressing the location concerning the center of dilation.

Given the focus on slopes and parallel lines, the best statements to support proving the dilation does not pass through the center of dilation are:

• "XY and X′Y′ both have an undefined slope with a center of dilation at (0,0). This proves that the lines are parallel and that the dilation does not pass through the center of dilation."
• "Modifying above upper X upper Y with bar and Modifying above upper X prime upper Y prime with bar both have an undefined slope with a center of dilation at (0,0). This proves that the lines are parallel and that the dilation does not pass through the center of dilation."

These statements adequately demonstrate that the dilation has resulted in parallel lines, indicating that the transformation did not occur through the designated center of dilation.

Answer 2

So out of all of the choices i gave you which one is it?

Answer 3

The correct choice to prove that the dilation of ΔXYZ does not pass through the center of dilation is:

"XY and X′Y′ both have an undefined slope with a center of dilation at (0,0). This proves that the lines are parallel and that the dilation does not pass through the center of dilation."

This statement clearly indicates that the lines are parallel due to having an undefined slope, which implies the dilation does not pass through the center of dilation at (0,0).