Dilation & Similarity — I can use dilations to define similarity and describe how lengths and angles change under dilation. - Math 2

SECTION 1

Historical Context & Motivation

Humans have been fascinated by the idea of scaling — making perfect copies of shapes at different sizes — for thousands of years. Ancient architects needed to enlarge blueprints into massive structures without distorting proportions, and mapmakers needed to shrink entire continents onto a single sheet of parchment. The mathematical concept of dilation formalizes this process, providing a precise transformation that scales figures up or down from a fixed center point. Understanding dilation is the key to defining what it truly means for two figures to be similar in geometry.

~300 BCE • Euclid's Elements

Euclid formalized the idea of similar figures in Book VI, proving that triangles with equal angles have proportional sides. This laid the groundwork for similarity theory.

~250 BCE • Archimedes & Scaling

Archimedes used scaling arguments to calculate areas and volumes of curved shapes, showing that proportional reasoning extends far beyond simple polygons.

1585 • Coordinate Geometry Emerges

With the development of coordinate systems, mathematicians could describe dilations algebraically — multiplying each coordinate by a constant factor — making transformations easier to compute.

1872 • Klein's Erlangen Program

Felix Klein proposed that geometry should be organized around transformations. Under this framework, similarity geometry is the study of properties preserved by dilations combined with rigid motions.

The central question that dilation answers is: How can we rigorously define what it means for two figures to have the same shape but different sizes? Congruence handles the case when two shapes are identical in every measurement, but similarity is a broader and equally powerful idea. Dilations give us the precise transformation that bridges the gap between congruence and similarity.

SECTION 2

Core Principles & Definitions

A dilation is a transformation that enlarges or shrinks a figure by a constant factor relative to a fixed point called the center of dilation. Unlike translations, rotations, or reflections — which are rigid motions that preserve both shape and size — a dilation preserves shape but changes size. The amount of scaling is controlled by the scale factor, often denoted k.

Center of Dilation

The fixed point from which all other points are pushed outward or pulled inward. Every ray from the center through a pre-image point also passes through the corresponding image point.

Scale Factor (k)

The ratio of a length in the image to the corresponding length in the pre-image. When k > 1 the figure enlarges; when 0 < k < 1 it shrinks.

Angle Preservation

Dilations do not change angle measures. Every angle in the image is congruent to the corresponding angle in the pre-image. This is why dilated figures look like the same shape.

Length Scaling

Every segment in the image has length equal to k times the corresponding segment in the pre-image. Ratios of corresponding sides are always equal to k.

Similarity Defined

Two figures are similar (△ABC ~ △DEF) if and only if one can be mapped onto the other by a sequence of rigid motions and a dilation. This is called a similarity transformation.

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KEY TAKEAWAY

Think of a dilation like using the zoom on a camera. When you zoom in, everything gets bigger, but a circle still looks like a circle, a square still looks like a square, and no angles become warped. The zoom level is your scale factor. Two figures are similar whenever you could zoom, slide, rotate, or flip one to land exactly on the other.

SECTION 3

Visualizing Dilation

The diagram below shows a triangle being dilated from a center point O with a scale factor of k = 2. Notice how each vertex of the image triangle is exactly twice as far from O as the corresponding vertex of the pre-image. The dashed rays from O through each vertex show this alignment.

The violet pre-image triangle △ABC is dilated from center O to produce the cyan image triangle △A'B'C'. Each image vertex lies on the ray from O through the pre-image vertex, at twice the distance. All angles are preserved; all side lengths are doubled.

Study the diagram carefully. The three dashed rays all originate at O and pass through corresponding vertex pairs (A and A', B and B', C and C'). Because k = 2, the distance OA' is exactly 2 × OA, and likewise for every other vertex. This proportional stretching is what guarantees that corresponding sides remain parallel and proportional, and that corresponding angles stay equal.

SECTION 4

Mathematical Framework

When a dilation is performed on the coordinate plane with center at the origin, the algebra is especially clean. Each point's coordinates are simply multiplied by the scale factor. For dilations centered at a point other than the origin, we translate first, scale, and translate back.

DILATION CENTERED AT THE ORIGIN

D_k(x, y) = (kx, ky)

Where k is the scale factor, and (x, y) is any point in the pre-image. The image point is (kx, ky).

DILATION CENTERED AT POINT (a, b)

D_k(x, y) = (a + k(x − a), b + k(y − b))

The center of dilation is (a, b). We subtract the center, multiply by k, then add the center back.

LENGTH RELATIONSHIP

A'B' = k × AB

Every segment in the image has length equal to the scale factor k times the corresponding segment in the pre-image.

ANGLE RELATIONSHIP

m∠A' = m∠A

All angle measures are preserved under dilation. This is the property that keeps the shape looking identical despite the size change.

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Important Note on Area

While lengths scale by k, areas scale by k². For example, if you dilate a triangle with scale factor 3, every side becomes 3× longer, but the area becomes 9× larger. This is because area is a two-dimensional measurement.

SECTION 5

Enlargements, Reductions & the Similarity Statement

The value of the scale factor k determines whether a dilation is an enlargement or a reduction. When k > 1, the image is larger than the pre-image (enlargement). When 0 < k < 1, the image is smaller (reduction). When k = 1, the image is congruent to the pre-image — no scaling happens at all. A negative scale factor reflects the figure through the center in addition to scaling.

Three dilations of the same square are compared. At k = ½ (cyan), the image shrinks. At k = 1 (amber), the figure is unchanged — a special case that gives congruence. At k = 2 (pink), the image doubles in size. In every case, angles remain 90°.

When we write a similarity statement like △ABC ~ △DEF, the order of vertices matters. It tells us that ∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F, and that the ratios AB/DE = BC/EF = AC/DF are all equal to the scale factor k. This proportionality of sides combined with congruence of angles is the hallmark of similar figures.

Scale Factor Range	Type	Effect on Lengths	Effect on Angles
0 < k < 1	Reduction	All lengths decrease (multiplied by k)	Unchanged
k = 1	Identity (Congruence)	All lengths stay the same	Unchanged
k > 1	Enlargement	All lengths increase (multiplied by k)	Unchanged

How the scale factor determines dilation type

SECTION 6

Worked Example: Dilating a Triangle on the Coordinate Plane

Let's work through a complete dilation problem step by step. Suppose triangle PQR has vertices P(2, 3), Q(6, 3), and R(4, 7). We want to dilate this triangle with center at the origin and scale factor k = 1.5, then verify that the image is similar to the original.

Dilating △PQR from the Origin with k = 1.5

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Step 1 — Identify the Given Information

Pre-image vertices: P(2, 3), Q(6, 3), R(4, 7). Center of dilation: origin (0, 0). Scale factor: k = 1.5. Since the center is at the origin, we use the formula D_k(x, y) = (kx, ky).
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Step 2 — Apply the Dilation to Each Vertex

Multiply each coordinate by 1.5: P' = (1.5 × 2, 1.5 × 3) = (3, 4.5) Q' = (1.5 × 6, 1.5 × 3) = (9, 4.5) R' = (1.5 × 4, 1.5 × 7) = (6, 10.5)

Image vertices: P'(3, 4.5), Q'(9, 4.5), R'(6, 10.5)
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Step 3 — Verify Length Scaling

Calculate PQ using the distance formula: PQ = √((6−2)² + (3−3)²) = √(16) = 4. Now P'Q' = √((9−3)² + (4.5−4.5)²) = √(36) = 6. Check the ratio: P'Q' / PQ = 6 / 4 = 1.5 ✓. This confirms that the side length scaled by k = 1.5.

P'Q' / PQ = 6 / 4 = 1.5 = k ✓
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Step 4 — Verify Angle Preservation

In △PQR, the base PQ is horizontal (both y-coordinates are 3). Calculate the slope of PR: (7−3)/(4−2) = 4/2 = 2. In △P'Q'R', the slope of P'R' = (10.5−4.5)/(6−3) = 6/3 = 2. Since corresponding sides have equal slopes (or are both horizontal), the angles between them are identical. The angles are preserved.

Corresponding angles are congruent ✓
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Step 5 — Write the Similarity Statement

Because the dilation maps △PQR to △P'Q'R' with all corresponding angles congruent and all corresponding sides in the ratio 1.5, we can confidently write the similarity statement.

△PQR ~ △P'Q'R' with scale factor k = 1.5

SECTION 7

Dilations vs. Rigid Motions

It's essential to understand how dilations differ from the rigid motions (translations, rotations, and reflections) that you studied earlier. Rigid motions produce congruent figures — same shape and same size. Dilations produce similar figures — same shape but possibly different size. Combining a dilation with one or more rigid motions gives a similarity transformation, which is the most general way to map one similar figure onto another.

Property	Rigid Motions	Dilation
Preserves angles?	Yes	Yes
Preserves side lengths?	Yes	No — multiplied by k
Preserves area?	Yes	No — multiplied by k²
Preserves parallelism?	Yes	Yes
Result	Congruent figures	Similar figures
Is it an isometry?	Yes (distances preserved)	Only if k = 1

Comparison of rigid motions and dilations

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KEY TAKEAWAY

Think of rigid motions as moving a photograph around on a table — you can slide it, spin it, or flip it over, but the picture itself stays the same size. A dilation is like printing that photograph at a different percentage — 50%, 150%, 200%. The content of the photo doesn't change, but the print size does. Congruence = same photo, same size. Similarity = same photo, any size.

SECTION 8

Connecting to Advanced Concepts

The ideas of dilation and similarity open the door to several powerful topics you'll encounter later in your math studies. The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar — no dilation computation needed. This shortcut works precisely because the dilation that would map one onto the other is guaranteed to exist.

Concept	This Lesson (Dilation & Similarity)	Advanced Extension
Proving similarity	Show a dilation + rigid motions map one figure to the other	AA, SAS~, SSS~ shortcut criteria
Scale factor	Ratio of image length to pre-image length	Ratios in similar solids (k for length, k² for area, k³ for volume)
Coordinate transformations	(x, y) → (kx, ky)	Matrix transformations and linear algebra
Proportional reasoning	Corresponding sides have equal ratios	Trigonometry — ratios that depend only on angle, not size

From dilation basics to advanced geometry

In trigonometry, you'll discover that the sine, cosine, and tangent of an angle are the same for every right triangle with that angle — regardless of the triangle's size. This works exactly because all such triangles are similar by AA, so the ratios of their sides are constant. The foundation for that powerful idea is the dilation concept you're learning right now.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Triangle DEF is dilated by a scale factor of 3 to form triangle D'E'F'. If ∠D = 50° and ∠E = 70°, what are the angle measures of △D'E'F'? Explain your reasoning.

Show answer

The angles of △D'E'F' are ∠D' = 50°, ∠E' = 70°, and ∠F' = 60°. Dilations preserve all angle measures — they only change lengths, not angles. Since the angles of a triangle sum to 180°, ∠F = 180° − 50° − 70° = 60°, and after dilation ∠F' = 60° as well. The scale factor of 3 affects side lengths (each is tripled) but has no impact on angles.

Question 2: BASIC CALCULATION

A rectangle has vertices A(1, 2), B(5, 2), C(5, 4), and D(1, 4). Dilate the rectangle from the origin with scale factor k = 2. Find the coordinates of each image vertex and the perimeter of the image rectangle.

Show answer

Multiply each coordinate by 2: A'(2, 4), B'(10, 4), C'(10, 8), D'(2, 8). The original rectangle has length AB = 4 and width BC = 2, so its perimeter is 2(4 + 2) = 12. In the image, A'B' = 8 and B'C' = 4, giving a perimeter of 2(8 + 4) = 24. Notice that the perimeter is exactly 2 × 12 = 24, confirming that perimeter (a one-dimensional measure) scales by the factor k = 2.

Question 3: INTERMEDIATE

Triangle JKL has vertices J(−2, 1), K(4, 1), and L(1, 5). It is dilated with center C(1, 1) and scale factor k = ⅓. Find the image vertices and show that the image triangle is similar to the original by verifying one pair of corresponding side lengths.

Show answer

Using

D_k

(x, y) = (a + k(x − a), b + k(y − b)) with center C(1, 1): J' = (1 + ⅓(−2 − 1), 1 + ⅓(1 − 1)) = (1 + (−1), 1 + 0) = (0, 1). K' = (1 + ⅓(4 − 1), 1 + ⅓(1 − 1)) = (1 + 1, 1 + 0) = (2, 1). L' = (1 + ⅓(1 − 1), 1 + ⅓(5 − 1)) = (1 + 0, 1 + 4/3) = (1, 7/3). Now verify: JK = |4 − (−2)| = 6, and J'K' = |2 − 0| = 2. The ratio J'K'/JK = 2/6 = ⅓ = k ✓. Since the dilation maps every length by the same factor and preserves all angles, △JKL ~ △J'K'L'.

Question 4: APPLIED

An architect creates a blueprint of a building where 1 cm on the blueprint represents 2.5 m in real life. On the blueprint, a rectangular room measures 4 cm by 3 cm. What are the actual dimensions of the room, what is the actual floor area, and what is the scale factor of the dilation from the blueprint to the real building?

Show answer

The scale factor from blueprint to real life is k = 2.5 m / 1 cm = 250 (converting to the same unit: 2.5 m = 250 cm, so k = 250). The actual length is 4 × 250 = 1000 cm = 10 m, and the actual width is 3 × 250 = 750 cm = 7.5 m. The blueprint area is 4 × 3 = 12 cm², and the actual area is 10 × 7.5 = 75 m². We can verify using the area scaling rule: 12 cm² × 250² = 12 × 62,500 = 750,000 cm² = 75 m² ✓. This confirms that area scales by k² while lengths scale by k.

Question 5: CRITICAL THINKING

A student claims: 'If I dilate a figure and then dilate the image again, the two dilations are equivalent to a single dilation.' Is this claim correct? If so, what is the scale factor of the combined dilation in terms of the individual scale factors k₁ and k₂? Justify your answer and determine whether the combined transformation is always a similarity transformation.

Show answer

The claim is correct. If you dilate by k₁ and then by k₂ (both from the same center), every length is first multiplied by k₁ and then by k₂, giving a net scale factor of k₁ × k₂. For example, a length L becomes k₁L after the first dilation, then k₂(k₁L) = (k₁k₂)L after the second. Angles are preserved by each dilation individually, so they are preserved by the composition. Therefore, the composition of two dilations from the same center is a single dilation with scale factor k₁ × k₂, and it is always a similarity transformation. Note: if the two dilations have different centers, the result may include a translation component, but the net effect on shape is still equivalent to a dilation (after accounting for position). The key insight is that the similarity relationship is transitive — if Figure A ~ Figure B and Figure B ~ Figure C, then Figure A ~ Figure C.

SUMMARY

Lesson Summary

Summary

A dilation is a transformation that scales a figure from a center of dilation by a constant scale factor k. On the coordinate plane with center at the origin, each point (x, y) maps to (kx, ky). When k > 1 the figure enlarges; when 0 < k < 1 it shrinks. In every dilation, angle measures are preserved and lengths are multiplied by k, while areas scale by k².

Two figures are similar if one can be mapped onto the other through a sequence of rigid motions (translations, rotations, reflections) combined with a dilation. Similar figures have congruent corresponding angles and proportional corresponding sides. This transformation-based definition of similarity connects directly to the AA, SAS~, and SSS~ criteria and forms the foundation for trigonometry, where ratios of sides in similar right triangles define sine, cosine, and tangent.