Geometric Transformations

🔁 Geometric Transformations

Part 1 of 5 — Moving Shapes & Translations

Topics in This Part

🔑 Key Concept: A transformation moves or changes a figure on the coordinate plane. In this lesson you'll master the four big ones — translations, reflections, rotations, and dilations — starting with the simplest: the slide.

What Is a Transformation?

A transformation takes an original figure (the pre-image) and produces a new figure (the image).

• The pre-image is what you start with. We label its points $A$, $B$, $C$, …
• The image is the result. We label its points with prime marks: $A'$, $B'$, $C'$ (read "A prime").

So if point $A$ moves to a new spot, that new spot is $A'$.

There are four transformations to know:

💡 The first three (slide, flip, turn) keep the figure exactly the same size and shape — only its position changes. We'll explore why that matters in Part 4.

Concept Check 🎯

Translations (Slides)

A translation slides every point of a figure the same distance in the same direction. Nothing flips, turns, or changes size.

On the coordinate plane we describe a slide by how it changes $x$ and $y$:

$(x,\, y) \to (x + a,\, y + b)$

• $a$ is the horizontal move: right is $+$, left is $-$.
• $b$ is the vertical move: up is $+$, down is $-$.

Worked Example: slide right 5, up 2

The rule is $(x, y) \to (x + 5,\, y + 2)$. Apply it to point $A(1, 3)$:

$A(1,\,3) \to A'(1 + 5,\; 3 + 2) = A'(6,\, 5)$

Worked Example: slide left 3, down 4

The rule is $(x, y) \to (x - 3,\, y - 4)$. Apply it to $B(2, 6)$:

$B(2,\,6) \to B'(2 - 3,\; 6 - 4) = B'(-1,\, 2)$

⚠️ Watch the signs. Left and down are negative. Mixing these up is the most common translation mistake.

Translate the Point 🧮

Apply the translation rule $(x, y) \to (x + 4,\, y - 3)$ (right 4, down 3) to each point. Enter the image coordinates.

1) $A(2, 5) \to A'(\,?,\ ?\,)$ — enter the $x$ then the $y$. 2) $B(-1, 0) \to B'(\,?,\ ?\,)$ — enter the $x$ then the $y$.

Name That Slide 🔽

A figure moves so that point $P(4, 1)$ lands on $P'(1, 5)$. Describe the translation.

🔁 Geometric Transformations

Part 2 of 5 — Reflections (Flips)

🔑 The Idea: A reflection flips a figure over a line called the line of reflection, producing a mirror image. The image is the same size and shape — just facing the other way.

Reflecting Over the Axes

When you flip a point over an axis, one coordinate keeps its sign and the other flips.

Why it works

Think of the $x$-axis as a horizontal mirror. A point $3$ units above it reflects to a point $3$ units below it — same $x$, opposite $y$.

Worked Example: reflect $A(4, 2)$ over the $x$-axis

Keep $x$, flip the sign of $y$:

$A(4,\, 2) \to A'(4,\, -2)$

Worked Example: reflect $B(-3, 5)$ over the $y$-axis

Flip the sign of $x$, keep $y$:

$B(-3,\, 5) \to B'(3,\, 5)$

💡 Memory trick: "Reflect over the $x$-axis → change the $y$." You change the coordinate of the axis you are not flipping over.

Concept Check 🎯

Two Special Lines: $y = x$ and the Origin

A couple of reflections come up often enough to memorize:

📝 Reflecting over the origin gives the same result as rotating $180^\circ$ about the origin — you'll see that again in Part 3.

Worked Example: reflect $A(2, 7)$ over the line $y = x$

Swap the coordinates:

$A(2,\, 7) \to A'(7,\, 2)$

Worked Example: reflect $B(-4, 6)$ over the origin

Flip both signs:

$B(-4,\, 6) \to B'(4,\, -6)$

Reflect the Point 🧮

Find each image. Enter the $x$-coordinate, then the $y$-coordinate.

1) Reflect $A(6, -2)$ over the $y$-axis $\to A'(\,?,\ ?\,)$ 2) Reflect $B(3, 8)$ over the line $y = x$ $\to B'(\,?,\ ?\,)$

Match the Reflection 🔽

Choose the correct image for each reflection of the point $(-5, 4)$.

🔁 Geometric Transformations

Part 3 of 5 — Rotations (Turns)

🔑 The Idea: A rotation turns a figure around a fixed point called the center of rotation. At this level the center is almost always the origin, and turns are counterclockwise unless told otherwise.

Rotations About the Origin

Memorize these three counterclockwise (CCW) rules:

Worked Example: rotate $A(3, 5)$ by $90^\circ$ CCW

Use $(x, y) \to (-y, x)$:

$A(3,\, 5) \to A'(-5,\, 3)$

Worked Example: rotate $B(-2, 6)$ by $180^\circ$

Use $(x, y) \to (-x, -y)$ — flip both signs:

$B(-2,\, 6) \to B'(2,\, -6)$

💡 Direction shortcut: A $270^\circ$ turn counterclockwise lands in the same place as a $90^\circ$ turn clockwise. Both use $(x, y) \to (y, -x)$.

⚠️ A $180^\circ$ rotation is the only one where the direction (CW vs. CCW) doesn't matter — you end up in the same spot either way.

Concept Check 🎯

Pick the Rule, Then the Image 🔽

You are rotating the point $P(2, 7)$ counterclockwise about the origin.

Rotate the Point 🧮

All rotations are counterclockwise about the origin. Enter the $x$-coordinate, then the $y$-coordinate.

1) Rotate $A(-3, 4)$ by $180^\circ \to A'(\,?,\ ?\,)$ 2) Rotate $B(6, 1)$ by $90^\circ \to B'(\,?,\ ?\,)$

🔁 Geometric Transformations

Part 4 of 5 — Dilations, Congruence & Similarity

🔑 The Big Distinction: Translations, reflections, and rotations are rigid motions — they preserve size and shape, so the image is congruent to the pre-image. A dilation resizes a figure, so the image is similar (same shape, different size).

Dilations (Resizing)

A dilation stretches or shrinks a figure from a center (usually the origin) by a scale factor $k$.

When the center is the origin, the rule is:

$(x,\, y) \to (kx,\, ky)$

The scale factor tells you what happens:

Worked Example: dilate $A(2, 3)$ by scale factor $k = 4$

Multiply both coordinates by $4$:

$A(2,\, 3) \to A'(8,\, 12)$

Worked Example: dilate $B(10, -6)$ by scale factor $k = \dfrac{1}{2}$

Multiply both coordinates by $\dfrac{1}{2}$:

$B(10,\, -6) \to B'(5,\, -3)$

⚠️ A dilation is the only one of the four transformations that changes the figure's size. The image is not congruent to the pre-image — it's similar.

Dilate the Point 🧮

Each dilation is centered at the origin. Enter the $x$-coordinate, then the $y$-coordinate.

1) Dilate $A(3, -5)$ by scale factor $k = 3 \to A'(\,?,\ ?\,)$ 2) Dilate $B(8, 12)$ by scale factor $k = \dfrac{1}{4} \to B'(\,?,\ ?\,)$

Congruent vs. Similar

This is the key idea Grade 8 transformations build toward:

🔑 Test for congruence: Two figures are congruent if you can map one onto the other using only rigid motions (slides, flips, turns) — no resizing.

🔑 Test for similarity: Two figures are similar if you can map one onto the other using rigid motions and a dilation.

Example

A triangle is translated 3 units right and then reflected over the $x$-axis. Since both moves are rigid motions, the image is congruent to the original. But if that triangle were also dilated by $k = 2$, the image would only be similar, not congruent.

Concept Check 🎯

Congruent or Similar? 🔽

For each sequence applied to a triangle, choose the strongest correct description of the image.

🔁 Geometric Transformations

Part 5 of 5 — Sequences & Mastery Check

You can now translate, reflect, rotate, and dilate any point. The last skill is applying two or more transformations in a row — a sequence — and then we'll finish with a mastery check.

Sequences of Transformations

To apply a sequence, transform the point with the first rule, then feed that result into the second rule. Order matters.

Worked Example

Start with $A(2, 3)$. Step 1 — reflect over the $x$-axis $(x, y) \to (x, -y)$:

$A(2,\, 3) \to A'(2,\, -3)$

Step 2 — translate right 4, up 1 $(x, y) \to (x + 4,\, y + 1)$, applied to $A'$:

$A'(2,\, -3) \to A''(6,\, -2)$

So the full sequence sends $A(2, 3) \to A''(6, -2)$. The double-prime $A''$ means "after two transformations."

⚠️ Order matters! Reflecting then sliding usually lands somewhere different from sliding then reflecting. Always work left to right, one step at a time.

💡 The image after a sequence of rigid motions is always congruent to the original — no matter how many slides, flips, and turns you string together.

Run the Sequence 🧮

Start with $P(1, 4)$. Apply the two steps in order.

Step 1: Rotate $90^\circ$ CCW about the origin, $(x, y) \to (-y, x)$. Step 2: Translate down 2, $(x, y) \to (x,\, y - 2)$.

Enter the final image $P''$ — the $x$-coordinate, then the $y$-coordinate.

Quick Reference

🔑 The first seven are rigid motions → image is congruent. A dilation (with $k \ne 1$) → image is similar.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.