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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

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 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.
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.

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 identify the option that explains how the lines corresponding to the segments of the triangles are not originating from the center of dilation.

The statement that best supports this is:

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 1/3.

However, this particular response does not directly suggest that the dilation does not pass through the center of dilation, so let's look at the other statement:

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 clearly demonstrates that since the lines are parallel (having the same undefined slope), they do not converge at the center of dilation, which is set at (0,0). Thus, this statement accurately proves that the dilation of \( \Delta XYZ \) does not pass through the center of dilation.

Therefore, the correct answer is the statement regarding their slopes being undefined, making them parallel and confirming that the dilation does not pass through the center.