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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
1
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 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⋅
1
3

,−3⋅
1
3

)=X′(−1,−1), Y′(−3⋅
1
3

,−6⋅
1
3

)=Y′(−1,−2), Z′(−6⋅
1
3

,−3⋅
1
3

)=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.

¯
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.
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Answer 1

The statement that proves that the dilation of \( \triangle XYZ \) does not pass through the center of dilation is:

"XY has a length of 3 and X′Y′ has a length of 1, which proves that \( \triangle X'Y'Z' \) is a dilation of \( \triangle XYZ \) by a scale factor of \( \frac{1}{3} \)."

This statement implies that the triangles are dilated but do not maintain the relationship to the center of dilation (which in this case is likely assumed to be the origin). If both lines have distinct lengths but are still proportionally related through a scale factor, that suggests that the dilation does not center at the origin.