To prove that the dilation of triangle XYZ does not pass through the center of dilation, we need to consider the properties of dilation in relation to the center of dilation.
A dilation centered at a point O with a scale factor k transforms a point P into a point P' such that:
\[ P' = O + k(P - O) \]
For a dilation to pass through the center of dilation, the image of a vertex of the triangle should align with the center point after the dilation transformation.
Let's denote the vertices of triangle XYZ:
- \( X(-3, -3) \)
- \( Y(-3, -6) \)
- \( Z(-6, -3) \)
We can find the centroid of triangle XYZ, which is the average of the vertices:
\[ \text{Centroid} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) = \left( \frac{-3 - 3 - 6}{3}, \frac{-3 - 6 - 3}{3} \right) = \left( \frac{-12}{3}, \frac{-12}{3} \right) = (-4, -4) \]
Assume that we apply a dilation with a center of dilation at some point O (not equal to the centroid) and a scale factor k. Let's denote the coordinates of the center of dilation as O(a, b).
To show that the dilation does not pass through O, we should show that the image points X', Y', Z' do not lie on the line through O. The line connecting O with the original points will not be consistent with the line to the dilated points unless O is on the line extending from any of the dilated images X', Y', Z'.
If we can find a specific scale factor and center that show the resulting points do not align with O, we can demonstrate the claim. For any arbitrary center O where O is not located at the centroid (-4, -4), we will prove that the transformation isn't centered by showing that the transformed coordinates do not remain collinear with O.
For example, if we assume a center of dilation, let's say O(0,0), when we perform a dilation of k=2, the images of the points under the transformation would be:
\[ X' = (0 + 2(-3 - 0), 0 + 2(-3 - 0)) = (-6, -6) \] \[ Y' = (0 + 2(-3 - 0), 0 + 2(-6 - 0)) = (-6, -12) \] \[ Z' = (0 + 2(-6 - 0), 0 + 2(-3 - 0)) = (-12, -6) \]
The resulting points (X', Y', Z') do not pass through the center O(0,0) or align in a linear fashion relative to O since they are different locations compared to the original triangle with their dilated position relative to O.
Thus, one strong statement would be:
"The dilation of triangle XYZ does not pass through the center of dilation (O) because the transformed points X', Y', and Z' do not align on the same line through O given that O is not equivalent to the centroid of triangle XYZ."
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