Dilations and Scale Factors

Resize from a center

A dilation tells every point how far to sit from one chosen point. Suppose a point $P$ is $4$ units from $O$. To scale from $O$ by a factor of $\frac 32$, place $P'$ on ray $OP$, $6$ units from $O$.

The mark $P'$ is read “$P$ prime.” It names the image, the point after the transformation; $P$ is the original point, or preimage. The ray fixes the direction, so the distance does not leave us choosing between two opposite sides of $O$.

A dilation with center $O$ and positive scale factor $k$ maps each point $P\ne O$ to the point $P'$ on ray $OP$ satisfying the rule below. The center itself maps to $O$.

The same center and factor apply to every point of a figure, not just its vertices. Join the images of the vertices to draw the image polygon.

• An enlargement has $k>1$. Every point other than $O$ moves farther from $O$.
• A reduction has $0<k<1$. Every point other than $O$ moves closer to $O$.
• The identity case has $k=1$. Every point stays where it is.

Zero and negative scale factors are outside this lesson. A fraction need not mean reduction; $\frac 54$ is greater than $1$.

Investigate what changes

A single factor appears to resize the whole figure without changing its angles. Test that claim in the figure. Blue solid edges show the original; red dashed edges show its image.

1. Keep the center still. Set the scale factor to $0.5$, then $1$, then $2$. Compare $AB$ with $A'B'$ and compare the two displayed angles.
2. Choose a factor other than $1$ and change each center coordinate. Follow the guide rays. Does each vertex image remain on its ray from the new center?
3. Watch the lines. Line $m$ contains $O$. Line $\ell$ does not. Compare each line with its image, and then test $k=1$ again.

At $k=2$, the $3$-unit side becomes $6$ units; at $k=0.5$, it becomes $1.5$ units. The angle opening stays the same. The away-center line shifts, while the line through the center still occupies the same straight path. With scripting unavailable, the displayed state shows one enlargement; the same comparisons can be checked with a ruler and protractor on a paper drawing.

Measurements support these patterns, but a few drawings cannot prove them for every figure. We explicitly accept the following dilation properties as foundations for the geometry that follows. We will use them to establish later results, rather than use those later results to prove them here.

• Lengths. For any distinct points $A,B$, their images satisfy $A'B'=k\cdot AB$.
• Angles. The angle between two rays equals the angle between their image rays.
• Lines. A line through $O$ maps onto the same line as a set. If $k\ne1$, a line away from $O$ maps onto a distinct parallel line. If $k=1$, every line maps onto itself.

These accepted properties will be the foundation for similarity in the next lesson.

Preserving an angle means preserving its measure, even when its vertex moves. Scaling a side means multiplying the distance between its endpoints, even when neither endpoint is the center.

Coordinates measure from the center

With the origin as center, multiply both coordinates by the scale factor. For example, a factor of $3$ sends $(2,-1)$ to $(6,-3)$. The horizontal displacement and the vertical displacement from the origin both triple, so the point stays in the same direction from the origin.

For a dilation centered at $(0, 0)$, the rule is

When the center is away from the origin, measure the displacements from that center first. Multiplying the raw coordinates would resize from the wrong point.

The same three moves work for any center $O(a,b)$. Subtract $a,b$ to measure from $O$, multiply both displacements by $k$, then add $a,b$ to return to the original coordinate system.

The coordinates $a,b$ locate the center; $k$ is the scale factor. The origin rule is the special case where both center coordinates are zero.

Fixed points and unchanged lines

A fixed point has its own location as its image. The center is always fixed because its distance from itself is zero. At scale factor $1$, all points are fixed.

For a different scale factor, a point away from the center cannot stay put. A point $5$ units from $O$, for example, moves to distance $10$ when $k=2$. In general, staying put would require both

Since $OP$ is positive away from the center, dividing by it forces $k=1$. Thus when $k\ne 1$, the center is the only fixed point.

A line can stay the same as a set while its points move. On a line through $O$, scaling sends points along its two opposite rays, and every location on that line is the image of some point. With $k=2$, a point $3$ units from $O$ moves to the location $6$ units from $O$; the original $3$-unit location is now occupied by the image of the point $1.5$ units from $O$. The entire line remains, but its individual points are not all fixed.

A stretch changes the shape

Making every horizontal distance twice as large while keeping vertical distances unchanged is a stretch in one direction. It uses different factors for different directions, so it is not a dilation of the plane.

Some angles can survive a stretch, such as the right angle between these horizontal and vertical legs. One unchanged angle does not establish that the whole transformation is a dilation. Check one common center, the ray condition, and one common factor for all points.

Common mistakes