Dilations & Scale Factor

Q: What happens when the scale factor is between 0 and 1?

The image shrinks. A scale factor of $\dfrac{1}{2}$ makes every length half as long, and every point lands halfway between the center and its original position. The rule $(x, y) \rightarrow (kx, ky)$ works exactly the same way — you just multiply by a fraction.

Q: Does a dilation change the angles of a figure?

No. A dilation multiplies every length by the scale factor but leaves every angle measure alone. That is why the image always has the same shape as the original — it is similar, just a different size.

Q: How is a dilation different from a translation or a rotation?

Translations, rotations, and reflections are rigid motions — they move a figure without changing any lengths, so the image is congruent. A dilation changes lengths, so the image is similar but usually not congruent. It is the one basic transformation that resizes.

Q: How do I find the scale factor from a graph?

Pick one vertex and its image, then divide a coordinate of the image by the matching coordinate of the original. If $B = (3, 1)$ and $B' = (6, 2)$, then $k = \dfrac{6}{3} = 2$. Check with the other coordinate: $\dfrac{2}{1} = 2$. Both should give the same $k$.

A dilation resizes a figure without changing its shape. Every point moves toward or away from a fixed center, and the amount of resizing is controlled by one number — the scale factor $k$ . If $k > 1$ the image is an enlargement; if $0 < k < 1$ the image is a reduction.

Dilations matter because they are the transformation behind similarity. Two figures are similar exactly when one is a dilation of the other (possibly combined with slides, flips, or turns). Master the coordinate rule below and every dilation problem becomes one multiplication.

The coordinate rule

When the center of dilation is the origin, the rule is as simple as it gets: multiply both coordinates by the scale factor. In symbols, $(x, y) \rightarrow (kx, ky)$ .

That is the entire computation. A point at $(2, 3)$ dilated by $k = 2$ lands at $(4, 6)$ . A point at $(6, -4)$ dilated by $k = \dfrac{1}{2}$ lands at $(3, -2)$ . Signs come along for the ride — a negative coordinate stays negative, it just gets scaled.

The graph below shows a triangle and its image under a dilation with scale factor $2$ centered at the origin. Each image vertex is exactly twice as far from the origin as the vertex it came from.

Finding the scale factor

If you are given a figure and its image, the scale factor is image divided by preimage: $k = \dfrac{\text{image length}}{\text{original length}}$ . Any pair of corresponding lengths works — a side, a coordinate, a distance from the center.

Keep the order straight. If a side of $4$ becomes a side of $10$ , then $k = \dfrac{10}{4} = 2.5$ . Flip the fraction and you get the scale factor for going the other direction, from the image back to the original.

What a dilation changes — and what it keeps

A dilation multiplies every length by $k$ , so sides, perimeters, and distances all scale by the same factor. But it never changes an angle. The image has exactly the same angle measures as the original.

That combination — same angles, proportional sides — is the definition of similar figures. So a dilation always produces a figure similar to the original, and it produces a congruent figure only in the special case $k = 1$ .

Worked examples

Example 1: dilate a point by $k = 2$

Dilate $A(2, 3)$ by scale factor $k = 2$ from the origin.

Write the coordinate rule

(x, y) \rightarrow (2x, 2y)

Multiply both coordinates by

2

(2 \cdot 2, \; 2 \cdot 3)

Simplify

A' = (4, 6)

Answer: $A' = (4, 6)$

Example 2: a reduction with $k = \dfrac{1}{2}$

Dilate $B(6, -4)$ by scale factor $k = \dfrac{1}{2}$ from the origin.

Write the coordinate rule

(x, y) \rightarrow \left(\tfrac{1}{2}x, \tfrac{1}{2}y\right)

Multiply both coordinates by

\dfrac{1}{2}

\left(\tfrac{1}{2} \cdot 6, \; \tfrac{1}{2} \cdot (-4)\right)

Simplify — the negative sign stays

B' = (3, -2)

Answer: $B' = (3, -2)$

Example 3: find the scale factor

A photo $4$ inches wide is enlarged so its width becomes $10$ inches. The original is $6$ inches tall. Find the scale factor and the new height.

Scale factor is image over original

k = \dfrac{10}{4} = 2.5

Apply

k

to the height

6 \cdot 2.5 = 15

State the new height

15 \text{ in}

Answer: $k = 2.5$ , so the enlarged photo is $15$ inches tall

Try one yourself

Common questions

What happens when the scale factor is between 0 and 1?

The image shrinks. A scale factor of $\dfrac{1}{2}$ makes every length half as long, and every point lands halfway between the center and its original position. The rule $(x, y) \rightarrow (kx, ky)$ works exactly the same way — you just multiply by a fraction.

Does a dilation change the angles of a figure?

No. A dilation multiplies every length by the scale factor but leaves every angle measure alone. That is why the image always has the same shape as the original — it is similar, just a different size.

How is a dilation different from a translation or a rotation?

Translations, rotations, and reflections are rigid motions — they move a figure without changing any lengths, so the image is congruent. A dilation changes lengths, so the image is similar but usually not congruent. It is the one basic transformation that resizes.

How do I find the scale factor from a graph?

Pick one vertex and its image, then divide a coordinate of the image by the matching coordinate of the original. If $B = (3, 1)$ and $B' = (6, 2)$ , then $k = \dfrac{6}{3} = 2$ . Check with the other coordinate: $\dfrac{2}{1} = 2$ . Both should give the same $k$ .

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Dilations & Scale Factor

The coordinate rule

Finding the scale factor

What a dilation changes — and what it keeps

Worked examples

Example 1: dilate a point by $k = 2$

Example 2: a reduction with $k = \dfrac{1}{2}$

Example 3: find the scale factor

Try one yourself

Common questions

Keep learning

Similar Polygons & Scale Factor

SSS & SAS Similarity

Similarity Proofs & Indirect Measurement

Dilations & Scale Factor

The coordinate rule

Finding the scale factor

What a dilation changes — and what it keeps

Worked examples

Example 1: dilate a point by k=2k = 2k=2

Example 2: a reduction with k=12k = \dfrac{1}{2}k=21​

Example 3: find the scale factor

Try one yourself

Common questions

Keep learning

Similar Polygons & Scale Factor

SSS & SAS Similarity

Similarity Proofs & Indirect Measurement

Example 1: dilate a point by $k = 2$

Example 2: a reduction with $k = \dfrac{1}{2}$