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which statement proves that the dilation of XYZ does not pass through the center of dilation? (X at the point (-3,-3) Y at (-3,-6) and Z at (-6,-3)

XY and X'Y' both have an undefined slope with a center of dialation at (0,0). This proves that the lines are parallel and that the dialation does not pass through the center of dilation.

X'(-3*1/3,13*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)

XY has a lenght of 3 and X'Y' has a lenght of 1, which proves that X'Y'Z' is a dilation of XYZ by a scale factor of 13.

response XY has a lenght of 3 and X'Y' has a length of 1, which proves that XYZ is a dilation of X'Y'Z' by a scale factor of 3.

1 answer

To determine whether the dilation of triangle XYZ passes through the center of dilation, we can analyze the statements provided.

  1. Understanding the Geometry: The points given are:

    • \( X(-3, -3) \)
    • \( Y(-3, -6) \)
    • \( Z(-6, -3) \)

  2. Center of Dilation: You mentioned that the assumed center of dilation is at (0,0).

  3. Slope Analysis: You noted that both segments \( XY \) and \( X'Y' \) have an undefined slope, indicating that these line segments are vertical. If these lines are parallel, it suggests that their corresponding dilated images (if the dilation were with respect to the origin) would also have to be vertical.

  4. Lengths and Scale Factor: You calculated the lengths:

    • Length of \( XY \) is 3 (found by subtracting the y-coordinates of Y and X).
    • Length of \( X'Y' \) is 1.

    The lengths confirm a scale factor of \( \frac{3}{1} = 3 \) when considering \( XYZ \) dilated to \( X'Y'Z' \). This establishes that the copy (XYZ) is larger than its dilated image (X'Y'Z'), supporting the idea of dilation, but it contradicts that the dilation is centered at (0,0).

  5. Proving It Does Not Pass Through the Center: The key statement that proves the dilation does not pass through the center of dilation is the observation about slopes and lengths. Since the segments \( XY \) and \( X'Y' \) are vertical and parallel (have undefined slopes), it indicates that there is no point through which both segments could pass if a dilation were centered at (0,0).

In conclusion, because:

  • The segments are vertical and do not intersect at (0,0).
  • The calculated lengths are consistent with a dilation factor but show proportional reduction rather than passing through the origin.

Thus, the statement about the slopes being undefined and the resultant shapes \( XY \) and \( X'Y' \) being parallel is a strong indicator that the dilation of triangle XYZ does not pass through the center of dilation at (0,0).