To determine which statement proves that the dilation of triangle XYZ does not pass through the center of dilation, we need to analyze the options provided.
Option A: "Line segment XY and line segment 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 is relevant because it refers to the slopes of the lines and the center of dilation at (0,0). If the lines are parallel and the dilation does not pass through the center of dilation, this indicates that the original and dilated triangles are not oriented in a way that aligns with point (0,0), negating the possibility of the dilation being centered there.
Option B: "Line segment XY has a length of 3 and line segment X'Y' has a length of 1, which proves that triangle XYZ is a dilation of triangle X'Y'Z' by a scale factor of 3."
While this statement establishes a relationship between the triangles, it does not provide evidence about whether the dilation passes through the center of dilation. It simply describes the scaling relationship.
Option C: "Line segment XY has a length of 3 and line segment X'Y' has length of , which proves that triangle X'Y'Z' is a dilation of triangle XYZ by a scale factor of 1/3."
This option also describes a relationship between the triangles but does not directly address the question of the center of dilation's position or whether the triangles retain proportional relationships when traced back to the center.
Conclusion: The best option that provides evidence regarding the center of dilation is Option A. It analyzes the slopes and parallel nature of the lines, drawing a conclusion about the center not aligning with the dilation, thus demonstrating that triangle XYZ does not pass through the center of dilation at (0,0).
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