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Dilations Keep
Lines Parallel

Discover how dilations transform lines in the plane: lines through the center stay put, while all others map to parallel images.

Geometry

What Is a Dilation?

A dilation is a transformation that enlarges or shrinks every point in the plane relative to a fixed point called the center of dilation. The amount of enlargement or shrinkage is controlled by the scale factor \( k \).

Center
The fixed point \( O \) from which all distances are measured.
Scale Factor
The ratio \( k \) by which every distance from \( O \) is multiplied.

Visualizing a Dilation

\(O\)
\(\ell\)
\(A\)
\(B\)
\(\ell'\)
\(A'\)
\(B'\)
\(k = 2\)
 ∥

Why Parallelism Matters

Dilations are the foundation of similarity in geometry. Two figures are similar when one can be mapped onto the other by a sequence of rigid motions and dilations. Parallelism under dilation guarantees that:

1
Angles are preserved: corresponding angles in similar figures remain equal.
2
Shape is preserved: figures keep their overall form even as size changes.
3
Proportionality holds: parallel image lines ensure side ratios stay constant.

The Core Theorem

This theorem has two cases depending on whether the original line passes through the center of dilation.

Case 1
Line does not pass through \( O \): the image is a distinct parallel line.
Case 2
Line passes through \( O \): the image is the same line (unchanged).

Case 1: Line Not Through the Center

\(O\)
\(\ell\)
\(P\)
\(Q\)
\(\ell'\)
\(P'\)
\(Q'\)
 ∥

Case 2: Line Through the Center

\(m\)
\(O\)
\(A\)
\(A'\)
\(B\)
\(B'\)
\(k = 2\)
\(k = 2\)
Image = Same Line \(m\)

Why Does This Work?

The proof relies on similar triangles formed by the dilation rays and the original line.

1
Pick two points \( P \) and \( Q \) on line \( \ell \) (which does not pass through \( O \)).
2
Form triangle \( \triangle OPQ \). Under dilation, \( P \to P' \) and \( Q \to Q' \) with \( \frac{OP'}{OP} = \frac{OQ'}{OQ} = k \).
3
By the Side-Splitter Theorem (or its converse), since the ratio is constant, \( P'Q' \parallel PQ \).
4
Since this holds for every pair of points on \( \ell \), the entire image line \( \ell' \) is parallel to \( \ell \).

Dilation vs. Translation

Both dilations and translations can produce parallel image lines, but they work very differently.

Dilation Translation
Type Non-rigid (changes size) Rigid (preserves size)
Center/Direction Fixed point \( O \) Fixed vector \( \vec{v} \)
Distances Multiplied by \( k \) Preserved exactly
Parallel images? Yes (if line does not pass through \( O \)) Always
Line through center Maps to itself Maps to parallel line

How to Apply the Theorem

1
Identify the center \( O \) and the scale factor \( k \).
2
Check: Does the line pass through \( O \)?
3
If NO: The image is a new parallel line at distance \( |k| \) times the original distance from \( O \).
4
If YES: The image is the same line. Points slide along it.
5
Verify: Map two specific points to confirm your conclusion.

Example 1: Coordinate Dilation

Dilation centered at the origin with \( k = 3 \). Line \( \ell \) passes through \( A(1,2) \) and \( B(3,2) \).

\(x\)
\(y\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(1\)
\(2\)
\(3\)
\(5\)
\(7\)
\(8\)
\(9\)
\(O\)
\(\ell\)
\(A(1,2)\)
\(B(3,2)\)
\(\ell'\)
\(A'(3,6)\)
\(B'(9,6)\)

Example 2: Sloped Line

Dilation centered at origin with \( k = 2 \). Line \( \ell \) passes through \( A(1,1) \) and \( B(3,2) \). Slope of \( \ell \): \( m = \frac{2-1}{3-1} = \frac{1}{2} \).

\(x\)
\(y\)
\(1\)
\(2\)
\(3\)
\(4\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(O\)
\(\ell\)
\(A(1,1)\)
\(B(3,2)\)
\(\ell'\)
\(A'(2,2)\)
\(B'(6,4)\)

Example 3: Line Through the Center

Dilation centered at origin, \( k = 2 \). Line \( m \) passes through \( O(0,0) \) and \( C(2,3) \). Does \( m \) map to a different line?

\(O\)
\(m\)
\(C(2,3)\)
\(C'(4,6)\)
Image = \(m\)
(same line)

Effect of Scale Factor \( k \)

The value of \( k \) determines the distance and direction of the image line relative to the center, but parallelism always holds.

\( k > 1 \)
Image line is farther from \( O \). Enlargement.
\( 0 < k < 1 \)
Image line is closer to \( O \). Reduction.
\( k < 0 \)
Image line is on the opposite side of \( O \). Still parallel.

Common Mistakes

✕ Forgetting Case 2
Claiming every line maps to a different parallel line. Lines through the center map to themselves.
✕ Confusing Parallel with Identical
Saying the image line is "the same" when the original line does not pass through the center.
✕ Ignoring Negative \( k \)
Assuming a negative scale factor changes the angle. It does not: the image is still parallel, just on the opposite side.

Summary at a Glance

Scenario Image of the Line Relationship
Line not through \( O \), \( k > 1 \) Farther from \( O \) Parallel
Line not through \( O \), \( 0 < k < 1 \) Closer to \( O \) Parallel
Line not through \( O \), \( k < 0 \) Opposite side of \( O \) Parallel
Line through \( O \), any \( k \) Same line Unchanged

Question 1 of 5

A dilation with center \( O \) and scale factor \( k = 3 \) is applied to line \( \ell \), which does not pass through \( O \). What is the image of \( \ell \)?

Question 2 of 5

Line \( m \) passes through the center of dilation \( O \). After a dilation with scale factor \( k = 5 \), what happens to \( m \)?

Question 3 of 5

A dilation centered at the origin maps \( A(2,4) \) to \( A'(6,12) \). Line \( \ell \) passes through \( A \) with slope \( \frac{3}{2} \). What is the slope of the image line \( \ell' \)?

Question 4 of 5

Which of the following best explains why a dilation maps a line not through the center to a parallel line?

Question 5 of 5

A dilation with center \( O \) and scale factor \( k = -2 \) is applied to line \( \ell \), which does not pass through \( O \). What is true about the image \( \ell' \)?