Geometry

Question Analysis  

The main focus is to determine the effect of dilation on the collinearity of points, using the property that dilations preserve collinearity.  

Key Concept Explanation  

Collinearity and Dilation: Dilations preserve collinearity. If points are collinear before dilation, they remain collinear after dilation, as dilations map straight lines to straight lines.  

Step-by-Step Solution  

1. Given Collinearity: Points , ,  are collinear.  

2. Dilation Effect: A dilation with scale factor  centered at the origin scales all coordinates but does not change the straight-line relationship between the points.  

Option Analysis  

Lie on a horizontal line: Incorrect. Collinearity does not imply horizontal alignment (original points may have any slope).  

Form a curved path: Incorrect. Dilations preserve straight lines, so points cannot form a curve.  

Remain collinear: Correct. Dilations preserve collinearity.  

Have no pattern: Incorrect. Collinear points will always follow a straight-line pattern after dilation.  

Common Mistakes  

Misassuming that a non-integer scale factor (