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Dilations &
Similarity

Learn how scale factors transform shapes, what makes figures similar, and how to connect dilations to real-world resizing.

Pre-Algebra

What Is a Dilation?

A dilation is a transformation that changes the size of a figure but keeps its shape exactly the same. Every dilation has two ingredients: a center of dilation (the fixed point) and a scale factor \( k \).

Center Original Dilated
\(k = 2\)

Understanding Scale Factor \( k \)

The scale factor \( k \) tells you how much every length in the figure is multiplied. It controls whether the image gets larger or smaller.

Enlargement
When \( k > 1 \), the image is larger than the original.
Reduction
When \( 0 < k < 1 \), the image is smaller than the original.
Congruent
When \( k = 1 \), the image is the same size: a congruent copy.

What Does Similar Mean?

Two figures are similar (written \( \triangle ABC \sim \triangle DEF \)) when they have the same shape but not necessarily the same size. Specifically:

1
All corresponding angles are equal. Each angle in one figure matches an angle in the other.
2
All corresponding sides are proportional. The ratios of matching sides are all the same value: the scale factor \( k \).

Dilations Produce Similar Figures

When you apply a dilation with scale factor \( k \) to any figure, the original and the image are always similar. Angles stay the same, and every side is multiplied by \( k \).

A B C
\(4\)
\(5\)
\(5\)
\(66^{\circ}\)
\(66^{\circ}\)
\(48^{\circ}\)
 ~ D E F
\(8\)
\(10\)
\(10\)
\(66^{\circ}\)
\(66^{\circ}\)
\(48^{\circ}\)
\(k = 2\)

The Similarity Rule

This means similarity is really about two steps:

1
Move it into position using translations, rotations, or reflections (these don't change size or shape).
2
Resize it with a dilation of scale factor \( k \) so all sides match.

Similar vs. Congruent

Property Similar (\(\sim\)) Congruent (\(\cong\))
Same shape ✓ Yes ✓ Yes
Same size ✕ Not necessarily ✓ Yes
Equal angles ✓ Yes ✓ Yes
Proportional sides ✓ Yes, ratio = \( k \) ✓ Yes, ratio = 1
Scale factor Any \( k > 0 \) \( k = 1 \)
Transformations Rigid motions + dilation Rigid motions only

How to Find the Scale Factor

1
Identify corresponding sides. Match each side in the original to the matching side in the image (same position, same angle pair).
2
Set up the ratio: $$ k = \frac{\text{image side}}{\text{original side}} $$
3
Simplify the fraction. The result is your scale factor.
4
Verify: Check that the same ratio works for all pairs of corresponding sides. If not, the figures are not similar.

Dilation on the Coordinate Plane

When the center of dilation is the origin \((0,0)\), every point \((x, y)\) maps to \((kx, ky)\). Simply multiply both coordinates by \( k \).

x y 1 2 3 4 5 6 7 1 2 3 4 5 6 O
\((1,1)\)
\((3,1)\)
\((2,3)\)
\((2,2)\)
\((6,2)\)
\((4,6)\)
\(k = 2\)

Example 1: Finding Scale Factor

Rectangle ABCD has sides \(6\) cm and \(4\) cm. Rectangle EFGH has sides \(15\) cm and \(10\) cm. Are they similar? If so, what is \( k \)?

A B C D
\(6\)
\(4\)
 E F G H
\(15\)
\(10\)
1
Match corresponding sides: \(6 \leftrightarrow 15\) and \(4 \leftrightarrow 10\).
2
Compute ratios: \(\frac{15}{6} = \frac{5}{2}\) and \(\frac{10}{4} = \frac{5}{2}\).
3
Ratios are equal, so the rectangles are similar with \( k = \frac{5}{2} = 2.5 \).

Example 2: Dilating a Triangle

\(\triangle PQR\) has vertices \(P(1, 2)\), \(Q(4, 2)\), \(R(1, 6)\). Dilate by \( k = \frac{1}{2} \) from the origin. Find the new vertices.

1
Multiply each coordinate by \( k = \frac{1}{2} \):
2
\( P'= \left(\frac{1}{2}, 1\right) \), \( Q'= \left(2, 1\right) \), \( R'= \left(\frac{1}{2}, 3\right) \).
3
Verify: Original base \( PQ = 3 \). New base \( P'Q' = 1.5 \). Ratio \( = \frac{1.5}{3} = \frac{1}{2} \) ✓

Example 3: Finding a Missing Side

\(\triangle ABC \sim \triangle DEF\). If \(AB = 8\), \(DE = 12\), and \(BC = 6\), find \(EF\).

A B C
\(8\)
\(6\)
 ~ D E F
\(12\)
?
1
Scale factor: \( k = \frac{DE}{AB} = \frac{12}{8} = \frac{3}{2} \).
2
Apply to BC: \( EF = k \times BC = \frac{3}{2} \times 6 = \) \(9\).

Similarity in the Real World

Dilations and similarity show up everywhere. Any time you resize something while keeping its proportions, you are working with similarity.

Maps
A map is a reduction of the real terrain. The map scale (e.g., \(1:50,000\)) is the scale factor.
Photos
Zooming a photo is an enlargement. The aspect ratio stays the same: similar rectangles.
Scale Models
Model cars and buildings are similar to the originals, with a known scale factor.

Solving with Proportions

When two figures are similar, you can set up a proportion to find unknown side lengths. Use cross-multiplication to solve.

1
Write the proportion with the unknown \(x\): \(\frac{8}{12} = \frac{x}{18}\).
2
Cross-multiply: \(8 \times 18 = 12 \times x\), so \(144 = 12x\).
3
Divide: \(x = \frac{144}{12} = \) \(12\).

Common Mistakes to Avoid

✕ Flipping the Ratio
Dividing original by image instead of image by original. This gives \( \frac{1}{k} \) instead of \( k \).
✕ Mismatching Sides
Comparing a side from one triangle to the wrong side of the other. Always match by position or angle.
✕ Adding Instead of Multiplying
Using \( k + \text{side} \) instead of \( k \times \text{side} \). Scale factor is a multiplier, not an addend.

Summary of Key Ideas

Concept What It Means Key Formula
Dilation Resize a figure from a center point \((x, y) \to (kx, ky)\)
Scale Factor \( k \) The multiplier for all lengths \( k = \frac{\text{image}}{\text{original}} \)
Similar (\(\sim\)) Same shape, proportional sides \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = k \)
Enlargement Image is bigger \( k > 1 \)
Reduction Image is smaller \( 0 < k < 1 \)

Question 1 of 5

A triangle with sides \(3, 4, 5\) is dilated by a scale factor of \( k = 4 \). What are the side lengths of the new triangle?

Question 2 of 5

Rectangle A is \(6\) by \(10\). Rectangle B is \(9\) by \(15\). What is the scale factor from A to B?

Question 3 of 5

Point \((6, 10)\) is dilated from the origin by \( k = \frac{1}{2} \). What are the new coordinates?

Question 4 of 5

\(\triangle ABC \sim \triangle XYZ\). If \( AB = 5 \), \( XY = 15 \), and \( YZ = 24 \), what is \( BC \)?

Question 5 of 5

Which statement is always true about two similar figures?