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All courses Math II · G-SRT.1.a 45 of 73
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.

Find a line image under dilation, including the invariant case

Problem
Find the exact image line under this dilation: center O=(0,0),k=2,line through P=(1,1),Q=(3,1).
Unworked dilation setup with center O(0,0), k=2, and source points P(1,1) and Q(3,1) on y=1; P′, Q′, and the image-line equation are not shown. Open full size
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Hint

Apply the coordinate rule to two distinct points on the line.

A dilation maps a center-passing line to itself; any other line maps to a parallel line.

Solution walkthrough

01

State the dilation rule

\[D_(O,2):(x,y)->(2x,2y)\]

A dilation centered at the origin multiplies both coordinates by the scale factor 2.

02

Map two distinct source points

\[P(1,1)->P'(2,2);~Q(3,1)->Q'(6,2)\]

The images of two different points determine the image line. Scaling both coordinates keeps each image on its ray from O.

03

Derive the image equation

\[P'_y=Q'_y=2~->~\text{image}~\text{line}~y=2\]

Both image points share y-coordinate 2 and have different x-coordinates, so their unique supporting line is horizontal at y=2.

04

Check the dilation theorem

\[\text{source}~y=1~\text{misses}~O;~\text{image}~y=2;~\text{slopes}~0~\text{and}~0;~\text{lines}~\text{distinct}\]

A line not through the dilation center maps to a distinct parallel line. The equations and equal slopes confirm that theorem here.

+

Another way

  1. Substitute y=y'/2 into source equation y=1, obtaining y'=2 directly.

!

Common mistake

Scale y as well as x. Mapping (1,1) to (2,1) is not a dilation by factor 2 from the origin.

Answer construction for center O(0,0) and k=2: P(1,1) maps to P′(2,2), and Q(3,1) maps to Q′(6,2). Thus source line y=1 maps to image line y=2, a distinct parallel line with the same direction.
Two transformed points determine the exact image line and its relation to the source.