Dilate a Point From the Origin

Dilating a point from the origin

A dilation centered at the origin changes a point’s distance from the origin by a scale factor, but keeps it on the same line through the origin. For a point ((x,y)), the image after dilation by factor (k) is ((kx,ky)).

Steps

1. Identify the original coordinates of the point.
2. Multiply both coordinates by the scale factor.
   • New point: ((kx,ky))
3. Simplify the result if needed.

Example pattern

If the point is ((3,-2)) and the scale factor is (4), then the dilated point is: [ (4\cdot 3,,4\cdot (-2))=(12,-8) ]

Quick check

• The new point should lie on the same ray from the origin as the original point.
• If the scale factor is greater than 1, the image is farther from the origin.
• If the scale factor is between 0 and 1, the image is closer to the origin.
• A negative scale factor places the image on the opposite side of the origin.

Use the coordinate multiplication rule carefully, and then verify the point’s position relative to the origin.