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Dilation

Dilation

A transformation in which a figure grows larger. Dilations may be with respect to a point (dilation of a geometric figure) or with respect to the axis of a graph (dilation of a graph).

Note: Some high school textbooks erroneously use the word dilation to refer to all transformations in which the figure changes size, whether the figure becomes larger or smaller. Unfortunately the English language has no word that refers collectively to both stretching and shrinking.

Pronunciation: Dilation (die-LAY-shun) has only three syllables, not four.

 

See also

Compression, compression of a geometric figure, compression of a graph

Key Formula

(x,y)→(k⋅x,  k⋅y)( x, y ) \rightarrow ( k \cdot x, \; k \cdot y )
Where:
  • (x,y)(x, y) = The coordinates of a point on the original figure
  • kk = The scale factor of the dilation (k > 1 for a true dilation)
  • (kx,ky)(kx, ky) = The coordinates of the corresponding point on the dilated image

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(3, 2), and B(2, 5). Dilate the triangle by a scale factor of 3 with center at the origin.
Step 1: Apply the dilation rule to point A by multiplying each coordinate by the scale factor k = 3.
A′=(3⋅1,  3⋅2)=(3,6)A' = (3 \cdot 1, \; 3 \cdot 2) = (3, 6)
Step 2: Apply the same rule to point B.
B′=(3⋅3,  3⋅2)=(9,6)B' = (3 \cdot 3, \; 3 \cdot 2) = (9, 6)
Step 3: Apply the rule to point C.
C′=(3⋅2,  3⋅5)=(6,15)C' = (3 \cdot 2, \; 3 \cdot 5) = (6, 15)
Step 4: Verify the dilation preserved the shape. The original base AB has length 2 and the new base A'B' has length 6, which is exactly 3 times as long — matching our scale factor.
A′B′AB=62=3=k\frac{A'B'}{AB} = \frac{6}{2} = 3 = k
Answer: The dilated triangle has vertices A'(3, 6), B'(9, 6), and C'(6, 15). Every side is 3 times as long as the original, and all angles are preserved.

Another Example

This example shows how to handle a dilation centered at a point other than the origin, which requires a translate–scale–translate-back approach.

Problem: A square has vertices P(2, 1), Q(4, 1), R(4, 3), and S(2, 3). Dilate the square by a scale factor of 2 with center at point P(2, 1), not the origin.
Step 1: When the center of dilation is not the origin, first translate each point so the center moves to the origin. Subtract the center coordinates (2, 1) from each point.
Pt=(0,0),Qt=(2,0),Rt=(2,2),St=(0,2)P_t = (0, 0), \quad Q_t = (2, 0), \quad R_t = (2, 2), \quad S_t = (0, 2)
Step 2: Multiply each translated coordinate by the scale factor k = 2.
Pt′=(0,0),Qt′=(4,0),Rt′=(4,4),St′=(0,4)P_t' = (0, 0), \quad Q_t' = (4, 0), \quad R_t' = (4, 4), \quad S_t' = (0, 4)
Step 3: Translate back by adding the center coordinates (2, 1) to each result.
P′=(2,1),Q′=(6,1),R′=(6,5),S′=(2,5)P' = (2, 1), \quad Q' = (6, 1), \quad R' = (6, 5), \quad S' = (2, 5)
Step 4: Check: the original square had side length 2. The dilated square has side length 4, which is 2 times the original — consistent with k = 2. Notice that P, the center of dilation, did not move.
Side length: 4=2×2  ✓\text{Side length: } 4 = 2 \times 2 \; \checkmark
Answer: The dilated square has vertices P'(2, 1), Q'(6, 1), R'(6, 5), and S'(2, 5). The center of dilation stays fixed.

Frequently Asked Questions

What is the difference between dilation and compression?
A dilation makes a figure larger (scale factor greater than 1), while a compression makes a figure smaller (scale factor between 0 and 1). Both transformations preserve the shape and angles of the figure, but they change its size in opposite directions. Many textbooks loosely use 'dilation' for both, but strictly speaking they are different.
Does a dilation change the shape of a figure?
No. A dilation preserves all angle measures and produces an image that is similar to the original. The sides all scale by the same factor, so the shape is identical — only the size changes. This is why dilations are classified as similarity transformations rather than rigid motions.
What happens when you dilate by a scale factor of 1?
A scale factor of 1 produces an image identical to the original — nothing changes. Technically this is the identity transformation. For a true dilation (enlargement), the scale factor must be greater than 1.

Dilation vs. Compression

DilationCompression
DefinitionA transformation that enlarges a figureA transformation that shrinks a figure
Scale factork > 10 < k < 1
Effect on distancesAll distances from the center increaseAll distances from the center decrease
Preserves shape?Yes — angles stay the sameYes — angles stay the same
Examplek = 3 triples all lengthsk = 1/3 reduces all lengths to one-third

Why It Matters

You encounter dilations throughout geometry when working with similar figures, scale drawings, and coordinate transformations. They are essential in real-world applications like map-making, architectural blueprints, and image resizing, where you need to enlarge a figure while keeping its proportions intact. On standardized tests and in courses from geometry through precalculus, dilation problems test your ability to apply scale factors and work with coordinate rules.

Common Mistakes

Mistake: Using the dilation formula (x, y) → (kx, ky) when the center is not the origin.
Correction: The simple multiplication rule only works when the center of dilation is at (0, 0). If the center is at point (a, b), you must first subtract (a, b), then multiply by k, then add (a, b) back: (x, y) → (k(x − a) + a, k(y − b) + b).
Mistake: Calling any size change a 'dilation,' even when the figure gets smaller.
Correction: Strictly, a dilation makes a figure larger (k > 1). When the figure shrinks (0 < k < 1), the correct term is compression. Using them interchangeably can lead to confusion on precise exam questions.

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