To determine whether the dilation of triangle XYZ to triangle X'Y'Z' passes through the center of dilation (which is given as (0,0)), we can analyze the properties of the slopes and lengths of the sides of the triangles.
The Key Statements:
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Slopes of XY and X'Y': If XY and X'Y' both have undefined slopes, it indicates that they are vertical lines. Since these lines are parallel (both vertical), the dilated segments do not intersect the origin because the center of dilation (0,0) does not lie on either line.
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Length Comparison: The calculation of the lengths of segments XY and X'Y' demonstrates a consistent scaling factor. If XY has a length of 3 and X'Y' has a length of 1, then the scaling factor is \( \frac{1}{3} \) (indicating that we are shrinking from XY to X'Y').
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Dilation Scale Factor: If you correctly state that XY and X'Y' are defined by these lengths, it reinforces that the triangles do not converge.
Conclusion:
Thus, the critical statement that proves the dilation 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 conclusion, in conjunction with the understanding of the slopes and lengths, shows that the transformation is consistent with dilations which do not pass through the center point (0,0).
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