Dilations & Similarity — I can use dilations and...

SECTION 1

Historical Context & Motivation

Have you ever zoomed in on a photo on your phone? The picture gets bigger, but everything stays in the right place. Noses don't suddenly stretch wider than faces! This idea — making things bigger or smaller while keeping the same shape — has been important to people for thousands of years.

Ancient builders, artists, and mapmakers all needed a way to take a real object and draw it at a different size. The math behind this process is called dilation (stretching or shrinking a figure), and the shapes it creates are called similar figures. Let's see how this idea developed over time.

~2500 BCE

Egyptian Pyramids

Ancient Egyptians used proportional drawings on grids to plan massive pyramids. Small sketches were scaled up to full-size buildings.

~300 BCE

Euclid's Elements

The Greek mathematician Euclid wrote formal rules about similar triangles and proportions. His ideas are still used in geometry today.

~1400s CE

Renaissance Art

Artists like Leonardo da Vinci used scale and proportion to make paintings look realistic. They enlarged small sketches into large murals.

Today

Digital Zooming & Maps

Every time you pinch-to-zoom on a screen or read a map scale, you are using dilation and similarity in everyday life.

The big question is: How do we describe exactly how much bigger or smaller a shape becomes, and how do we know two shapes are truly similar? That is what this lesson will answer.

SECTION 2

Core Principles & Definitions

Before we dive into examples, let's lock in the key vocabulary. These four ideas are the building blocks for everything else in the lesson.

Dilation

A dilation is a transformation (a change) that makes a shape bigger or smaller. The shape keeps its angles and proportions.

Scale Factor

The scale factor (often written as k) is the number you multiply each length by. If k > 1, the shape grows. If 0 < k < 1, the shape shrinks.

Center of Dilation

The center of dilation is the fixed point from which the shape stretches or shrinks. Think of it like an anchor.

Similar Figures

Two figures are similar (symbol: ~) if one can be turned into the other using dilations and other transformations like slides, flips, or turns.

✦ KEY TAKEAWAY

Think of a dilation like a copy machine with a zoom setting. If you set the zoom to 200%, every length doubles — the paper gets bigger, but the picture still looks the same. The zoom percentage is like the scale factor. A 200% zoom means a scale factor of 2. A 50% zoom means a scale factor of 0.5.

SECTION 3

Visual Explanation — Seeing Dilation in Action

The best way to understand dilation is to see it. In the diagram below, a small triangle is dilated from a center point. Notice how every vertex (corner) moves away from the center by the same scale factor.

The blue triangle (original) is dilated by a scale factor of 2 from the red center point. Every side of the purple triangle (image) is exactly twice as long. The dashed lines show how each vertex moves outward from the center.

Look at the dashed lines in the diagram. They go from the center of dilation through each vertex of the original triangle and keep going to the matching vertex of the new triangle. The distance from the center to A' is exactly twice the distance from the center to A. That ratio is the scale factor, k = 2.

💡 Quick Check

If the scale factor were k = 3, every side would be three times as long. If k = 0.5, every side would be half as long. The angles always stay the same!

SECTION 4

Mathematical Framework

Now let's put the ideas into formulas you can use. There are two main equations to know.

SCALE FACTOR

k = New Length ÷ Original Length

k = scale factor. If k > 1, the figure gets larger (an enlargement). If 0 < k < 1, the figure gets smaller (a reduction). If k = 1, nothing changes.

IMAGE COORDINATES

(x', y') = (k × x, k × y)

When the center of dilation is the origin (0, 0), multiply each coordinate by k to find the new point. For example, if A = (3, 4) and k = 2, then A' = (6, 8).

SIMILARITY RULE

△ABC ~ △A'B'C' when AB/A'B' = BC/B'C' = AC/A'C' and ∠A = ∠A', ∠B = ∠B', ∠C = ∠C'

Two figures are similar when all matching sides share the same ratio (the scale factor) and all matching angles are equal.

📐 REMEMBER

Congruent shapes are like identical twins — same size and shape. Similar shapes are like a parent and child — they look alike, but one is bigger. The scale factor tells you the "growth factor" between them.

SECTION 5

Types of Dilations — Enlargements vs. Reductions

Dilations come in two flavors: enlargements and reductions. The diagram below shows both side by side on a coordinate grid so you can compare them.

Left: An enlargement with k = 2 doubles every side length. Right: A reduction with k = 0.5 cuts every side in half. Both images are similar to the original because the angles stay the same.

Comparing enlargements and reductions

Feature	Enlargement (k > 1)	Reduction (0 < k < 1)
Size of image	Bigger than the original	Smaller than the original
Side lengths	Multiplied by k (lengths increase)	Multiplied by k (lengths decrease)
Angle measures	Stay exactly the same	Stay exactly the same
Similar to original?	Yes	Yes

SECTION 6

Worked Example — Finding the Scale Factor and New Coordinates

Let's work through a full problem step by step. Read each step carefully and make sure you understand it before moving on.

Dilating a Triangle on the Coordinate Plane

Step 1 — Read the ProblemTriangle PQR has vertices P(2, 3), Q(6, 3), and R(4, 7). It is dilated from the origin with a scale factor of k = 3. Find the vertices of the image triangle P'Q'R' and verify the triangles are similar.

Step 2 — Apply the Scale Factor to Each VertexMultiply each coordinate by k = 3. P' = (3 × 2, 3 × 3) = (6, 9) Q' = (3 × 6, 3 × 3) = (18, 9) R' = (3 × 4, 3 × 7) = (12, 21)

P'(6, 9), Q'(18, 9), R'(12, 21)

Step 3 — Check a Side Length RatioOriginal side PQ = 6 − 2 = 4 units (horizontal line). Image side P'Q' = 18 − 6 = 12 units. Ratio: 12 ÷ 4 = 3. That matches k = 3. ✓

Side ratio = 3 = k ✓

Step 4 — Verify SimilaritySince we used a dilation (which keeps all angles equal and multiplies all sides by the same factor), the two triangles are similar. We write △PQR ~ △P'Q'R' with scale factor 3.

△PQR ~ △P'Q'R', k = 3

SECTION 7

Similar vs. Congruent — What's the Difference?

Students sometimes mix up similar and congruent. The table below makes the differences crystal clear.

Similar vs. Congruent Figures

Feature	Similar (~)	Congruent (≅)
Same shape?	Yes	Yes
Same size?	Not necessarily	Yes, always
Angles	All matching angles are equal	All matching angles are equal
Sides	Proportional (same ratio)	Exactly the same length
Scale factor	Any positive number	Exactly 1
Transformations used	Dilation + slides, flips, turns	Only slides, flips, turns (no dilation)

✦ KEY TAKEAWAY

Think of it this way: congruent figures are like two identical slices of pizza from the same pie. Similar figures are like a personal pizza and a large pizza — same shape, different sizes. All congruent figures are also similar (with k = 1), but not all similar figures are congruent.

SECTION 8

Connection to Advanced Geometry

The ideas you learned today are the foundation for much of high-school geometry. Here is a preview of where similarity goes next.

From middle school to high school geometry

What You Learned Today	Where It Leads
Scale factor k multiplies each side	In high school, you'll prove triangle similarity using AA, SAS, and SSS similarity theorems
Matching angles stay equal	This becomes the Angle-Angle (AA) postulate for proving triangles similar
Dilations on the coordinate plane	Leads to composition of transformations and matrix operations
Using ratios to compare sides	Connects to trigonometry — sine, cosine, and tangent are ratios in similar right triangles

You will also see similarity in science classes. For example, engineers build scale models of bridges and test them before building the real thing. The model and the real bridge are similar figures connected by a scale factor!

SECTION 9

Practice Problems

Try these five problems on your own. Start with the first one and work your way up. Check your answer after each problem.

PROBLEM 1 — CONCEPTUAL

A triangle is dilated with a scale factor of k = 4. What happens to its angles? What happens to its side lengths?

PROBLEM 2 — BASIC CALCULATION

A rectangle has a width of 5 cm and a length of 12 cm. It is dilated by a scale factor of k = 3. What are the width and length of the new rectangle?

PROBLEM 3 — INTERMEDIATE

Point A(4, 6) is dilated from the origin to create A'(10, 15). What is the scale factor k?

PROBLEM 4 — APPLIED

A map uses a scale of 1 cm = 50 km. Two cities are 7 cm apart on the map. What is the actual distance between them? What is the scale factor from the map to the real world?

PROBLEM 5 — CRITICAL THINKING

Triangle DEF has sides 6, 8, and 10. Triangle GHI has sides 9, 12, and 15. Are these triangles similar? If so, what transformations would map one onto the other? Could they also be congruent?

SUMMARY

Lesson Summary

A dilation is a transformation that changes the size of a figure without changing its shape. It is defined by a center of dilation (the anchor point) and a scale factor k (the multiplier for every length). When k > 1, the figure gets bigger (enlargement). When 0 < k < 1, it gets smaller (reduction). On the coordinate plane with center at the origin, you find new points using (x', y') = (k × x, k × y).

Two figures are similar if one can be mapped onto the other by a dilation (and possibly slides, flips, or turns). Similar figures have equal angles and proportional sides. Remember: congruent figures are a special case of similar figures where k = 1. Scale factor, dilations, and similarity will be key tools in high-school geometry and beyond!