Which statement proves that the dilation of ΔXYZ

Question

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.

Answer 1

The correct statement that 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 indicates that both segments (in the original triangle and the dilated triangle) are parallel and that they do not intersect the center of dilation, confirming that the dilation does not pass through the center.