Question
Question
A coordinate plane's axes range from negative 8 to 0, both by 1-unit increments. Triangle upper X (-3,-3) upper Y (-3,-6) upper Z (-6,-3) and triangle upper X prime (-1,-1) upper Y prime (-1,-2) upper Z prime (-2,-1) 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¯¯¯¯¯¯¯¯ 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. X′(−3⋅13,−3⋅13)=X′(−1,−1), Y′(−3⋅13,−6⋅13)=Y′(−1,−2), Z′(−6⋅13,−3⋅13)=Z′(−2,−1)
A coordinate plane's axes range from negative 8 to 0, both by 1-unit increments. Triangle upper X (-3,-3) upper Y (-3,-6) upper Z (-6,-3) and triangle upper X prime (-1,-1) upper Y prime (-1,-2) upper Z prime (-2,-1) 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¯¯¯¯¯¯¯¯ 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. X′(−3⋅13,−3⋅13)=X′(−1,−1), Y′(−3⋅13,−6⋅13)=Y′(−1,−2), Z′(−6⋅13,−3⋅13)=Z′(−2,−1)
Answers
GPT-4o mini
To determine which statement proves that the dilation of triangle ΔXYZ does not pass through the center of dilation, we need to analyze the characteristics of dilation, particularly in relation to the center of dilation and the scale factor involved.
The two triangles are defined as:
- Triangle ΔXYZ with vertices at X(-3, -3), Y(-3, -6), Z(-6, -3)
- Triangle ΔX'Y'Z' with vertices at X'(-1, -1), Y'(-1, -2), Z'(-2, -1)
We find the lengths of the segments based on the points provided:
- The length of segment XY is √[(-3 - (-3))^2 + (-3 - (-6))^2] = √(0 + 9) = 3.
- The length of segment X'Y' is √[(-1 - (-1))^2 + (-1 - (-2))^2] = √(0 + 1) = 1.
Now let's analyze the options given in the responses.
### Statements Analysis:
1. **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.**
This indicates a scale factor, but it does not explain why the dilation does not pass through the center.
2. **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 connects the undefined slopes to the conclusion that the dilation does not pass through (0,0), providing a reasoning.
3. **XY̅ 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.**
This is incorrect; it's not proof of the dilation not passing through the center.
4. **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)**
This notation calculates the coordinates but does not directly imply the position of the center of dilation as being passed through or not.
### Conclusion:
The 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 highlights the key reason why the dilation relationship does not pass through the center.
The two triangles are defined as:
- Triangle ΔXYZ with vertices at X(-3, -3), Y(-3, -6), Z(-6, -3)
- Triangle ΔX'Y'Z' with vertices at X'(-1, -1), Y'(-1, -2), Z'(-2, -1)
We find the lengths of the segments based on the points provided:
- The length of segment XY is √[(-3 - (-3))^2 + (-3 - (-6))^2] = √(0 + 9) = 3.
- The length of segment X'Y' is √[(-1 - (-1))^2 + (-1 - (-2))^2] = √(0 + 1) = 1.
Now let's analyze the options given in the responses.
### Statements Analysis:
1. **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.**
This indicates a scale factor, but it does not explain why the dilation does not pass through the center.
2. **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 connects the undefined slopes to the conclusion that the dilation does not pass through (0,0), providing a reasoning.
3. **XY̅ 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.**
This is incorrect; it's not proof of the dilation not passing through the center.
4. **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)**
This notation calculates the coordinates but does not directly imply the position of the center of dilation as being passed through or not.
### Conclusion:
The 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 highlights the key reason why the dilation relationship does not pass through the center.