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Dilations Change Length
by Scale Factor

Discover how every dilation multiplies segment lengths by a constant ratio, preserving shape while changing size.

Geometry

What Is a Dilation?

A dilation is a transformation that enlarges or reduces a figure by a constant multiplier called the scale factor, denoted \( k \). Every point moves along a ray from a fixed center of dilation, and all distances are multiplied by \( k \).

Enlargement
When \( k > 1 \), the image is larger than the original.
Reduction
When \( 0 < k < 1 \), the image is smaller than the original.

Visualizing a Dilation

\( O \)
\( A \)
\( B \)
\( 4 \text{ cm} \)
\( A' \)
\( B' \)
\( 8 \text{ cm} \)
\( k = 2 \) (enlargement)

Why Scale Factor Matters

The scale factor \( k \) is the single number that controls every length in a dilated figure. If you know \( k \), you can predict any new measurement from the original: just multiply.

Architecture
Blueprints are dilations of real buildings. A 1:100 scale means \( k = \frac{1}{100} \).
Maps
Every map legend describes a dilation scale factor between paper and reality.
Photography
Zooming a camera lens applies a dilation to the image on the sensor.

The Dilation Length Theorem

This means the ratio of the image length to the original length is always equal to the scale factor: $$ \frac{A'B'}{AB} = k $$

Theorem in Action

\( P \)
\( Q \)
\( R \)
\( 3 \)
\( 4 \)
\( 5 \)
\( P' \)
\( Q' \)
\( R' \)
\( 6 \)
\( 8 \)
\( 10 \)
→
\( k = 2 \)

Dilation vs. Other Transformations

Dilation Rigid Motion
Lengths change? Yes: multiplied by \( k \) No: stay the same
Angles change? No: preserved No: preserved
Shape preserved? Yes (similar) Yes (congruent)
Size preserved? No (unless \( k=1 \)) Yes
Examples Scale drawing, zoom Translation, rotation, reflection

How to Apply the Theorem

1
Identify the scale factor \( k \). It may be given directly, or you can compute it from one known pair of corresponding lengths.
2
Find the original length. Measure or read the pre-image segment you need.
3
Multiply: \( \text{Image length} = k \times \text{Original length} \).
4
Check direction: If \( k > 1 \), the image is longer. If \( 0 < k < 1 \), the image is shorter.
5
Verify with a second pair of corresponding segments to confirm your scale factor.

Example 1: Finding the Image Length

A segment \( \overline{MN} \) has length 6 cm. A dilation with scale factor \( k = 3 \) is applied. What is \( M'N' \)?

\( M \)
\( N \)
\( 6 \text{ cm} \)
→ \( k = 3 \)
\( M' \)
\( N' \)
\( 18 \text{ cm} \)
Pre-image
Image

Example 2: Finding the Scale Factor

A segment \( \overline{AB} \) has length 10 cm. After a dilation, the image \( \overline{A'B'} \) has length 4 cm. What is the scale factor?

\( A \)
\( B \)
\( 10 \text{ cm} \)
→ \( k = ? \)
\( A' \)
\( B' \)
\( 4 \text{ cm} \)
Pre-image
Image

Example 3: Full Triangle Dilation

Triangle \( \triangle DEF \) has sides \( DE = 5 \), \( EF = 12 \), \( DF = 13 \). A dilation with \( k = \frac{1}{2} \) is applied. Find all image side lengths.

\( D \)
\( E \)
\( F \)
\( 5 \)
\( 12 \)
\( 13 \)
→
\( k = \frac{1}{2} \)
\( D' \)
\( E' \)
\( F' \)
\( 2.5 \)
\( 6 \)
\( 6.5 \)

Working Backwards: Finding the Original

Sometimes you know the image length and the scale factor, and you need the original. Rearrange the formula:

Example: After a dilation with \( k = 4 \), a segment measures 20 cm. The original was: $$ \frac{20}{4} = 5 \text{ cm} $$

Dilation on the Coordinate Plane

When the center of dilation is the origin, every coordinate is multiplied by \( k \):

\( x \)
\( y \)
\( O \)
\( 2 \)
\( 4 \)
\( 6 \)
\( 2 \)
\( 4 \)
\( A(1,1) \)
\( B(3,2) \)
\( A'(2,2) \)
\( B'(6,4) \)
\( k = 2 \)

Common Mistakes to Avoid

✕ Adding Instead of Multiplying
Students write \( A'B' = AB + k \) instead of \( A'B' = k \cdot AB \). Dilation is multiplicative, not additive.
✕ Inverting the Ratio
Computing \( k = \frac{AB}{A'B'} \) instead of \( k = \frac{A'B'}{AB} \). The image goes in the numerator.
✕ Forgetting Fractions
When \( k = \frac{3}{4} \), some students round to 1 and conclude lengths don't change. Always compute the full product.

Summary: Scale Factor at a Glance

Scale Factor \( k \) Effect on Length Type
\( k > 1 \) Image is longer Enlargement
\( k = 1 \) Image is same length Identity (no change)
\( 0 < k < 1 \) Image is shorter Reduction

Question 1 of 5

A segment has length 9 cm. After a dilation with scale factor \( k = \frac{2}{3} \), what is the image length?

Question 2 of 5

A segment \( \overline{PQ} \) is 5 cm long. Its image \( \overline{P'Q'} \) after a dilation is 20 cm. What is the scale factor?

Question 3 of 5

Triangle \( \triangle ABC \) has sides 6, 8, and 10. It is dilated by \( k = 2.5 \). What is the longest side of the image?

Question 4 of 5

After a dilation centered at the origin, point \( A(3, 4) \) maps to \( A'(9, 12) \). What is the image of segment \( \overline{BC} \) if \( BC = 7 \)?

Question 5 of 5

A dilation maps \( \overline{XY} \) (length 16) to \( \overline{X'Y'} \) (length 12). Which statement is true?