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

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To determine which statement proves that the dilation of ΔXYZ does not pass through the center of dilation, we need to focus on the properties of dilation and the relationship between the original triangle and its image.

The most relevant statement is as follows:

"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 effectively demonstrates that the lines representing the original segment XY and the dilated segment X'Y' are parallel due to both having undefined slopes, indicating they are vertical lines. Since both lines are vertical and the center of dilation is at (0, 0), this shows that the dilation does not pass through the center of dilation because the image is not along the line connecting (0,0) to both points, but rather distinct from the center.

The other options focus on lengths and scale factors, which do not directly prove the position relationship relative to the center of dilation. Therefore, the first statement is indeed the correct response.

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

1 answer

To determine which statement proves that the dilation of ΔXYZ does not pass through the center of dilation, we need to focus on the properties of dilation and the relationship between the original triangle and its image.

The most relevant statement is as follows:

"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 effectively demonstrates that the lines representing the original segment XY and the dilated segment X'Y' are parallel due to both having undefined slopes, indicating they are vertical lines. Since both lines are vertical and the center of dilation is at (0, 0), this shows that the dilation does not pass through the center of dilation because the image is not along the line connecting (0,0) to both points, but rather distinct from the center.

The other options focus on lengths and scale factors, which do not directly prove the position relationship relative to the center of dilation. Therefore, the first statement is indeed the correct response.

Ask a New Question or answer this question.

GPT-4o mini · Answer

To determine which statement proves that the dilation of ΔXYZ does not pass through the center of dilation, we need to focus on the properties of dilation and the relationship between the original triangle and its image.

The most relevant statement is as follows:

"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 effectively demonstrates that the lines representing the original segment XY and the dilated segment X'Y' are parallel due to both having undefined slopes, indicating they are vertical lines. Since both lines are vertical and the center of dilation is at (0, 0), this shows that the dilation does not pass through the center of dilation because the image is not along the line connecting (0,0) to both points, but rather distinct from the center.

The other options focus on lengths and scale factors, which do not directly prove the position relationship relative to the center of dilation. Therefore, the first statement is indeed the correct response.