Rotations and Dilations

🔄 Rotations & Dilations

Part 1 of 5 — Rotations: Turning Around a Point

Topics in This Part

🔑 Key Concept: A rotation turns every point of a figure around a fixed center by the same angle. The figure keeps its exact size and shape — it just faces a new direction. This is the first of two transformations in this lesson; the second is dilation, which resizes a figure.

What Is a Rotation?

Imagine pinning a photo to a corkboard with a single tack and spinning it. Every point on the photo travels along a circle around that tack. That is a rotation.

A rotation is defined by three things:

🔑 Key Idea: A rotation is a rigid motion (also called an isometry). The image is congruent to the original — same side lengths, same angles, same area. Only the orientation in space changes.

The original figure is called the pre-image. The turned figure is the image, and we mark its points with a prime symbol: $A \to A'$ (read "$A$ prime").

Concept Check 🎯

Rotating Around the Origin

When the center is the origin $(0,0)$, the coordinate rules are clean and worth memorizing. For a counterclockwise rotation:

Worked Example: rotate $A(4, 2)$ counterclockwise by $90°$

Apply $(x, y) \to (-y, x)$ with $x = 4$ and $y = 2$:

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

The point moved from the lower-right region toward the upper-left — exactly what a quarter-turn counterclockwise should do.

💡 Shortcut to remember: A $270°$ counterclockwise turn lands in the same place as a $90°$ clockwise turn. So "$90°$ clockwise" uses the rule $(x, y) \to (y, -x)$.

Rotate 90° Counterclockwise 🧮

Use the rule $(x, y) \to (-y, x)$. Enter each image coordinate.

1) $P(3, 5) \to P'(\,?,\ ?\,)$ — enter the $x$-coordinate, then the $y$-coordinate. 2) $Q(-1, 6) \to Q'(\,?,\ ?\,)$ — enter the $x$-coordinate, then the $y$-coordinate.

Concept Check 🎯

🔄 Rotations & Dilations

Part 2 of 5 — Mastering Rotation Rules

🔑 The Goal: In Part 1 you met the three origin rotations. Now you'll apply them fluently, handle clockwise turns, and rotate whole figures point by point.

Clockwise vs. Counterclockwise

Every counterclockwise (CCW) rotation has a matching clockwise (CW) one that lands in the same place:

So you only ever need the three CCW rules:

⚠️ Watch out: "$90°$ clockwise" is not the $90°$ CCW rule. It equals $270°$ CCW, so it uses $(x, y) \to (y, -x)$. Read the direction carefully before you pick a rule.

Match the Rule 🔽

Choose the correct coordinate rule for each rotation about the origin.

Rotating a Whole Figure

To rotate a polygon, rotate each vertex with the rule, then reconnect them in order.

Worked Example: rotate triangle $RST$ by $180°$ about the origin

Vertices: $R(2, 1)$, $S(5, 1)$, $T(2, 4)$. Apply $(x, y) \to (-x, -y)$:

$R(2, 1) \to R'(-2, -1)$ $S(5, 1) \to S'(-5, -1)$ $T(2, 4) \to T'(-2, -4)$

Connect $R'S'T'$ in the same order. The new triangle is congruent to the original and sits in the opposite quadrant.

✅ Check: Side $RS$ had length $5 - 2 = 3$. Side $R'S'$ runs from $(-2,-1)$ to $(-5,-1)$, length $\lvert -5 - (-2) \rvert = 3$. ✓ Congruence preserved.

Rotate the Vertices 🧮

Rotate each point $270°$ counterclockwise about the origin using $(x, y) \to (y, -x)$.

1) $M(2, 7) \to M'(\,?,\ ?\,)$ — $x$ then $y$. 2) $N(-4, 3) \to N'(\,?,\ ?\,)$ — $x$ then $y$.

Concept Check 🎯

🔄 Rotations & Dilations

Part 3 of 5 — Dilations: Resizing a Figure

🔑 A New Kind of Transformation: Rotations keep size fixed. A dilation does the opposite job — it stretches or shrinks a figure while keeping its shape. The result is similar (same shape, proportional size), not congruent.

What Is a Dilation?

A dilation resizes a figure away from (or toward) a fixed center of dilation by a number called the scale factor, written $k$.

$\text{dilation about origin:}\quad (x, y) \to (kx,\ ky)$

Every coordinate is multiplied by $k$. The scale factor tells you what happens:

🔑 Key Idea: A dilation produces a figure that is similar to the original. All angles stay equal, and every length is multiplied by $k$. Only when $k = 1$ is the image also congruent.

⚠️ Don't confuse the two! A rotation keeps size and changes orientation. A dilation keeps orientation and changes size.

Concept Check 🎯

Dilating Points and Figures (center = origin)

Multiply every coordinate by the scale factor $k$.

Worked Example: dilate $D(3, -6)$ by $k = 2$

$D(3, -6) \to D'(2 \cdot 3,\ 2 \cdot (-6)) = D'(6, -12)$

The point moved twice as far from the origin in every direction.

Worked Example: dilate a triangle by $k = \frac{1}{2}$

Vertices $A(4, 2)$, $B(8, 6)$, $C(2, -4)$ with $k = \frac{1}{2}$:

$A(4, 2) \to A'(2, 1)$ $B(8, 6) \to B'(4, 3)$ $C(2, -4) \to C'(1, -2)$

Each coordinate was halved, so the image triangle is half the size — a reduction.

💡 Distance from the center scales by $k$ too. If a vertex was $10$ units from the origin and $k = \frac{1}{2}$, the image vertex sits $5$ units from the origin along the same ray.

Apply the Scale Factor 🧮

Dilate each point about the origin. Multiply both coordinates by $k$.

1) $P(5, 3)$ with $k = 4 \to P'(\,?,\ ?\,)$ — $x$ then $y$. 2) $Q(-8, 12)$ with $k = \frac{1}{4} \to Q'(\,?,\ ?\,)$ — $x$ then $y$.

Concept Check 🎯

🔄 Rotations & Dilations

Part 4 of 5 — Comparing, Combining & Working Backward

🔑 Putting It Together: You can now rotate and dilate. This part sharpens the difference between them, finds an unknown scale factor or angle, and chains transformations into a single sequence.

Rotation vs. Dilation at a Glance

💡 Both transformations preserve angle measures and keep the figure's overall shape. The dividing line is size: a rotation never resizes, while a dilation (with $k \ne 1$) always does.

Rotation or Dilation? 🔽

For each described change, choose whether it is a rotation or a dilation, and whether the image is congruent or similar.

Working Backward to Find the Transformation

Sometimes you are given the pre-image and image and must identify the transformation.

Find the rotation: $A(1, 4) \to A'(-4, 1)$

Test the rules. Does $(x, y) \to (-y, x)$ fit? $(-4, 1)$ would need $-y = -4$ (so $y = 4$ ✓) and $x = 1$ ✓. It's a $90°$ counterclockwise rotation.

Find the scale factor: $B(2, 5) \to B'(8, 20)$

Divide matching coordinates: $\frac{8}{2} = 4$ and $\frac{20}{5} = 4$. It's a dilation with $k = 4$.

⚠️ Always test BOTH coordinates. A real dilation gives the same $k$ from $x$ and from $y$. If the two ratios disagree, it is not a single dilation about the origin.

Find the Missing Value 🧮

1) Dilation about the origin: $(6, -9) \to (2, -3)$. What is the scale factor $k$? (fraction or decimal) 2) Dilation about the origin with $k = 5$ sends $(x, y) \to (35, -10)$. Find the original $x$, then the original $y$.

Combining Two Transformations

You can apply one transformation, then another, to the result. Order matters — but doing the steps carefully keeps it simple.

Worked Example: rotate $90°$ CCW, then dilate by $k = 2$

Start with $P(3, 1)$.

Step 1 — rotate $90°$ CCW with $(x, y) \to (-y, x)$: $P(3, 1) \to P'(-1, 3)$

Step 2 — dilate $P'$ by $k = 2$ with $(x, y) \to (2x, 2y)$: $P'(-1, 3) \to P''(-2, 6)$

The final image $P''(-2, 6)$ is both turned a quarter-turn and twice as far from the origin.

💡 A rotation-then-dilation (or the reverse) is called a similarity transformation: the final image is always similar to the original, because only the dilation step changes size.

Concept Check 🎯

🔄 Rotations & Dilations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) rotate $90°$, $180°$, and $270°$ about the origin, (2) handle clockwise turns, (3) dilate by any scale factor, (4) tell rotations from dilations, and (5) work backward to find an angle or scale factor. Let's bring it all together.

Quick Reference

⚠️ Remember the big contrast: a rotation keeps size and changes facing; a dilation keeps facing and changes size. Both keep the shape and all angle measures.

Mixed Practice 🔽

Fill in each result.

Apply It 🧮

A triangle has a vertex at $V(10, -5)$.

1) After a dilation about the origin with $k = \frac{2}{5}$, the image vertex is $V'(\,?,\ ?\,)$ — enter $x$ then $y$. 2) Starting again from $V(10, -5)$, after a $180°$ rotation about the origin, the image vertex is $V''(\,?,\ ?\,)$ — enter $x$ then $y$.

Exit Quiz ✅

Answer all three to finish the lesson.