Understand Similarity Through Transformations

SECTION 1

Historical Context & Motivation

People have noticed similar shapes for thousands of years. Ancient builders, artists, and scientists all needed to copy shapes at different sizes. Think about making a bigger version of a map or shrinking a blueprint. The math behind similarity helped solve those real-world problems.

~300 BC

Euclid's Elements

The Greek mathematician Euclid wrote a book of geometry rules. He described similar figures as shapes with equal angles and sides in the same ratio.

~200 BC

Eratosthenes Measures the Earth

Eratosthenes used similar triangles and shadows to estimate the circumference of the Earth — a powerful real-world use of similarity.

1800s

Transformations Formalized

Mathematicians like Felix Klein began describing geometry using transformations — movements like slides, flips, turns, and resizing. This gave us a new way to define similarity.

2010

Common Core Standards

The Common Core (CCSS) defined similarity using transformations. Two figures are similar if one can be turned into the other by rotations, reflections, translations, and dilations.

So here is the big question this lesson answers: How can we prove two shapes are similar by describing the exact moves and resizing needed to turn one into the other?

SECTION 2

Core Principles & Definitions

Before we dive in, let's make sure you know the key vocabulary. There are four types of transformations you need to understand. The first three — translations, reflections, and rotations — are called rigid motions because they move a figure without changing its size or shape. The fourth, dilation, changes the size but keeps the same shape.

Translation (Slide)

Sliding a figure up, down, left, or right without turning or flipping it. Every point moves the same distance in the same direction.

Reflection (Flip)

Flipping a figure over a line (called the line of reflection). It creates a mirror image.

Rotation (Turn)

Spinning a figure around a fixed point (called the center of rotation) by a certain number of degrees.

Dilation (Resize)

Enlarging or shrinking a figure from a center point by a scale factor. A scale factor of 2 doubles the size. A scale factor of ½ halves it.

Similar Figures

Two figures are similar if you can get from one to the other using a sequence of translations, reflections, rotations, and dilations.

✦ KEY TAKEAWAY

Think of similarity like a photo on your phone. You can slide the photo around the screen (translate), flip it (reflect), rotate it, and pinch to zoom in or out (dilate). After all those moves, the picture still looks the same — just in a different position or size. Similar figures are the same shape, but not necessarily the same size.

SECTION 3

Visual Explanation — Seeing Similarity in Action

The diagram below shows two similar triangles on a coordinate plane. Triangle ABC (the smaller one in blue) can be transformed into Triangle A′B′C′ (the larger one in pink) using a specific sequence of transformations. Let's see how.

Triangle ABC (blue) has vertices at A(1,1), B(2,1), and C(1,2). Triangle A′B′C′ (pink) has vertices at A′(3,2), B′(5,2), and C′(3,4). The pink triangle is twice the size of the blue triangle and has been slid to the right and up. A dilation by scale factor 2 followed by a translation maps one onto the other.

Notice how both triangles are right triangles with the same angle measures. The sides of Triangle A′B′C′ are exactly 2 times longer than the matching sides of Triangle ABC. That factor of 2 is the scale factor of the dilation. Because we can get from one triangle to the other using a dilation and a translation, the two triangles are similar.

SECTION 4

Mathematical Framework — How Transformations Work

Each transformation has a mathematical rule. You can describe each move with coordinates. Let's look at the rules one at a time.

TRANSLATION

(x, y) → (x + a, y + b)

Where a is the number of units you slide left/right, and b is the number of units you slide up/down.

REFLECTION OVER THE Y-AXIS

(x, y) → (−x, y)

The x-coordinate flips sign (positive becomes negative, or vice versa). The y-coordinate stays the same.

ROTATION 90° COUNTERCLOCKWISE ABOUT THE ORIGIN

(x, y) → (−y, x)

The coordinates swap, and the new x-coordinate gets a negative sign. Other rotation amounts have different rules.

DILATION FROM THE ORIGIN

(x, y) → (k × x, k × y)

Where k is the scale factor. If k > 1 the figure gets bigger. If 0 < k < 1 the figure gets smaller.

💡 Important Distinction

Translations, reflections, and rotations produce congruent figures (same size AND shape). A dilation changes size, so when we include a dilation the figures are similar (same shape but possibly different size).

SECTION 5

Describing Sequences of Transformations

To show two figures are similar, you need to describe the exact sequence of transformations that maps one onto the other. Think of it like giving step-by-step directions. The diagram below shows a figure going through three transformations in order.

This flowchart shows a triangle going through three transformations in order: first a dilation by a scale factor of 2, then a reflection, and finally a translation. The final green triangle is similar to the original blue triangle.

The order of transformations matters. You might dilate first and then translate, or you might translate first and then dilate. Both can work, but the specific numbers in each step will change depending on the order you choose.

Step 1 — Identify corresponding vertices (matching corners) between the two figures.
Step 2 — Calculate the scale factor by dividing a side length of the new figure by the matching side length of the original.
Step 3 — Decide which rigid motions (translation, reflection, rotation) you need to line up the figures after dilating.
Step 4 — Write out the complete sequence in order.

SECTION 6

Worked Example — Mapping One Rectangle to Another

Rectangle PQRS has vertices P(0, 0), Q(4, 0), R(4, 2), and S(0, 2). Rectangle P′Q′R′S′ has vertices P′(−3, 1), Q′(−3, 7), R′(−6, 7), and S′(−6, 1). Describe a sequence of transformations that shows these rectangles are similar.

Finding the Transformation Sequence

Step 1 — Find the Side LengthsRectangle PQRS has length PQ = 4 and width QR = 2. Rectangle P′Q′R′S′ has one pair of sides of length 6 (from P′ to Q′: |7 − 1| = 6) and another pair of length 3 (from Q′ to R′: |−6 − (−3)| = 3).

PQRS: 4 × 2. P′Q′R′S′: 6 × 3.

Step 2 — Find the Scale FactorCompare matching sides. The longer side of PQRS is 4 and the longer side of P′Q′R′S′ is 6. The scale factor k = 6 ÷ 4 = 1.5. Check the shorter sides: 3 ÷ 2 = 1.5. Both ratios match!

Scale factor k = 1.5

Step 3 — Apply the DilationDilate PQRS by a scale factor of 1.5 from the origin. P(0,0) → (0,0). Q(4,0) → (6,0). R(4,2) → (6,3). S(0,2) → (0,3). Now the rectangle is 6 × 3.

After dilation: (0,0), (6,0), (6,3), (0,3)

Step 4 — Identify the Needed Rigid MotionOur dilated rectangle sits in the first quadrant. But P′Q′R′S′ is in the second quadrant and the long side is vertical, not horizontal. Notice the long side switched from horizontal to vertical. A 90° counterclockwise rotation about the origin will do this. Applying (x, y) → (−y, x): (0,0)→(0,0), (6,0)→(0,6), (6,3)→(−3,6), (0,3)→(−3,0).

After rotation: (0,0), (0,6), (−3,6), (−3,0)

Step 5 — Translate to Final PositionCompare our rotated rectangle (0,0), (0,6), (−3,6), (−3,0) with the target P′(−3,1), Q′(−3,7), R′(−6,7), S′(−6,1). We need to slide every point 3 left and 1 up: (x, y) → (x − 3, y + 1). Check: (0,0) → (−3,1) ✓, (0,6) → (−3,7) ✓, (−3,6) → (−6,7) ✓, (−3,0) → (−6,1) ✓.

Sequence: Dilate by 1.5, rotate 90° CCW, translate (−3, +1). The rectangles are similar.

SECTION 7

Similar vs. Congruent — What's the Difference?

Students sometimes mix up similar and congruent. They are related, but they are not the same. The table below highlights the key differences.

Comparing congruent and similar figures

Feature	Congruent Figures	Similar Figures
Shape	Same	Same
Size	Same	Can be different
Angles	All corresponding angles equal	All corresponding angles equal
Side lengths	All corresponding sides equal	Corresponding sides in the same ratio
Transformations needed	Translations, reflections, rotations only	Translations, reflections, rotations, AND dilations
Scale factor	Always 1	Can be any positive number

✦ KEY TAKEAWAY

All congruent figures are also similar (with a scale factor of 1). But not all similar figures are congruent. It's like how all squares are rectangles, but not all rectangles are squares. Similarity is the bigger category that includes congruence as a special case.

SECTION 8

Connection to Advanced Geometry

Understanding similarity through transformations sets you up for important ideas in high school geometry and beyond. Here is how this concept connects to what comes next.

How today's concepts lead to future learning

What You Learn Now (8th Grade)	What Comes Next (High School & Beyond)
Describe similarity using a sequence of transformations	Prove triangles are similar using AA, SAS, and SSS similarity theorems
Use scale factors to compare side lengths	Use similarity to solve for missing sides and set up proportions
Understand that dilations change size but keep shape	Apply similarity to real-world problems like indirect measurement and scale models
Combine rigid motions with dilations	Use coordinate geometry proofs involving transformations

The transformation approach to similarity is also the foundation of trigonometry. When you study sine, cosine, and tangent, you are using ratios of sides in similar right triangles. So the work you are doing now will pay off for years to come!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Triangle DEF is translated 5 units to the right and then reflected over the x-axis. Is the resulting triangle congruent to, similar to, or neither compared to the original? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

A triangle has vertices at A(2, 3), B(6, 3), and C(2, 7). You dilate it from the origin with a scale factor of 3. What are the coordinates of A′, B′, and C′?

PROBLEM 3 — INTERMEDIATE

Rectangle JKLM has vertices J(1, 1), K(5, 1), L(5, 3), and M(1, 3). Rectangle J′K′L′M′ has vertices J′(2, 2), K′(10, 2), L′(10, 6), and M′(2, 6). Describe a sequence of transformations that maps JKLM to J′K′L′M′. What is the scale factor?

PROBLEM 4 — APPLIED

An architect draws a floor plan where 1 cm on the plan equals 2 meters in real life. A room on the plan is a rectangle measuring 3 cm by 4 cm. The actual room is 6 m by 8 m. Explain how the plan and the actual room are related using the language of similarity and transformations.

PROBLEM 5 — CRITICAL THINKING

Triangle PQR has vertices P(0, 0), Q(3, 0), and R(0, 4). Triangle P′Q′R′ has vertices P′(4, 2), Q′(4, −4), and R′(−4/3, 2). Could these triangles be similar? If so, describe a possible sequence of transformations. If not, explain why.

SUMMARY

Lesson Summary

Two figures are similar if you can map one onto the other using a sequence of translations, reflections, rotations, and dilations. The first three are rigid motions that keep size and shape the same. A dilation changes size using a scale factor while keeping the shape the same.

To describe the similarity between two figures, identify the scale factor by comparing corresponding side lengths, then determine which rigid motions (slides, flips, turns) are needed to line up the figures after resizing. Congruent figures are a special case of similarity where the scale factor equals 1. Remember: same shape = similar; same shape AND same size = congruent.