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Question

A coordinate plane's axes range from negative 8 to 0, both by 1-unit increments. Triangle upper X upper Y upper Z and triangle upper X prime upper Y prime upper Z prime are plotted.

Which statement proves that the dilation of ΔXYZ does not pass through the center of dilation?

(1 point)
Responses

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¯¯¯¯¯¯¯¯ 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.

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¯¯¯¯¯¯¯¯ 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.

Answer 1

To determine which statement proves that the dilation of triangle \( \Delta XYZ \) does not pass through the center of dilation, we need to look for a response that indicates the relationship between the two triangles and the center of dilation based on their coordinates or slopes.

The correct choice 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 option effectively demonstrates that if the segments are parallel and do not intersect at the center of dilation, it indicates that the dilation does not pass through that point.