Which statement proves that the dilation of ΔXYZ does not pass through the center of dilation? (1 point) Responses 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 . 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 . 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.

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The statement that best proves 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 highlights that both the original segment \( XY \) and its image \( X'Y' \) have an undefined slope, indicating that they are vertical lines and thus parallel. Since the dilation does not intersect the center of dilation \((0,0)\), it confirms that the dilation does not pass through this center.

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Which statement proves that the dilation of ΔXYZ does not pass through the center of dilation? (1 point) Responses 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 . 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 . 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.

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The statement that best proves 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 highlights that both the original segment \( XY \) and its image \( X'Y' \) have an undefined slope, indicating that they are vertical lines and thus parallel. Since the dilation does not intersect the center of dilation \((0,0)\), it confirms that the dilation does not pass through this center.

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GPT-4o mini · Answer

The statement that best proves 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 highlights that both the original segment \( XY \) and its image \( X'Y' \) have an undefined slope, indicating that they are vertical lines and thus parallel. Since the dilation does not intersect the center of dilation \((0,0)\), it confirms that the dilation does not pass through this center.