Transformations — translation, reflection, rotation, enlargement (basic)

Overview

Transformations are operations that move, flip, turn, or resize shapes while preserving certain properties. They're the foundation of symmetry, graphics, and understanding how objects relate in space.

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Core Concepts

[!intuition] Why Transformations Matter

Think of transformations as instructions for moving a shape. Imagine you have a photo: translation slides it to a new position, reflection flips it like a mirror, rotation spins it around a point, and enlargement makes it bigger or smaller. Each transformation has rules that determine what stays the same (like angles or shape) and what changes (like position or size).

Understanding transformations lets you:

• Predict where shapes will land after movement
• Recognize patterns and symmetry
• Navigate coordinate geometry
• Build intuition for vectors and matrices later

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[!definition] The Four Basic Transformations

1. Translation: Sliding every point of a shape the same distance in the same direction

   • Described by a vector $\begin{pmatrix} a \\ b \end{pmatrix}$ where $a$ = horizontal shift, $b$ = vertical shift
   • Shape, size, and orientation unchanged

2. Reflection: Flipping a shape over a mirror line

   • Each point maps to its mirror image equidistant from the line
   • Shape and size unchanged, orientation reversed

3. Rotation: Turning a shape around a fixed center by an angle

   • Needs: center point, angle (°), direction (clockwise/anticlockwise)
   • Shape and size unchanged, position orientation change

4. Enlargement: Scaling a shape from a center by a scale factor

   • Scale factor $k > 1$ → bigger; $0 < k < 1$ → smaller; $k = 1$ → same size
   • Shape unchanged, size changes (unless $k=1$), position changes (unless center is on the shape)

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Derivations & Formulas

[!formula] Translation Vector

What it does: Moves point $(x, y)$ by adding shifts

If we translate by vector $\begin{pmatrix} a \\ b \end{pmatrix}$:

$\text{New point: } (x', y') = (x + a, y + b)$

Why this formula?

• Horizontal shift: $a$ units right (positive) or left (negative)
• Vertical shift: $b$ units up (positive) or down (negative)
• Every point moves the same amount → shape slides as a rigid unit

Derivation from scratch: Translation means "every point moves by the same displacement." If the displacement is $a$ horizontally and $b$ vertically, then:

• Original point: $(x, y)$
• Add horizontal displacement: $x \to x + a$
• Add vertical displacement: $y \to y + b$
• Result: $(x + a, y + b)$

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[!formula] Reflection Across x-axis

What it does: Flips points over the $x$-axis (horizontal line $y=0$)

$\text{Point } (x, y) \to (x, -y)$

Why this works?

• The $x$-coordinate stays the same (no horizontal movement)
• The $y$-coordinate flips sign (point moves to opposite side of axis)
• Distance from axis preserved: if point is $d$ units above, image is $d$ units below

Derivation: The $x$-axis is the line $y = 0$. To reflect $(x, y)$:

1. The perpendicular distance from $(x, y)$ to the axis is $|y|$
2. The reflected point must be the same distance on the opposite side
3. If original $y > 0$ (above axis), reflected $y'< 0$ (below axis) and $|y'| = |y|$
4. Therefore $y' = -y$, and $x' = x$ (no horizontal shift)

Other common reflections:

• y-axis $(y=0)$: $(x, y) \to (-x, y)$
• Line $y=x$: $(x, y) \to (y, x)$ — swap coordinates
• Line $y=-x$: $(x, y) \to (-y, -x)$ — swap and negate

Why $y=x$ swaps coordinates? The line $y=x$ is diagonal. A point $(3, 1)$ is 2 units below the line; its reflection $(1, 3)$ is 2 units above. Geometrically, reflecting across $y=x$ exchanges the roles of $x$ and $y$.

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[!formula] Rotation Around Origin

What it does: Spins point $(x, y)$ around $(0,0)$ by angle $\theta$ anticlockwise

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$

Expanded: $x' = x\cos\theta - y\sin\theta$ $y' = x\sin\theta + y\cos\theta$

Why this formula?

Think of $(x,y)$ in polar coordinates: distance $r = \sqrt{x^2+y^2}$ from origin, angle $\phi$ from positive $x$-axis.

1. Original: $x = r\cos\phi$, $y = r\sin\phi$
2. After rotating by $\theta$, new angle is $\phi + \theta$
3. New coordinates: $x' = r\cos(\phi+\theta) = r(\cos\phi\cos\theta - \sin\phi\sin\theta) = x\cos\theta - y\sin\theta$ $y' = r\sin(\phi+\theta) = r(\sin\phi\cos\theta + \cos\phi\sin\theta) = y\cos\theta + x\sin\theta$

Common angles:

• 90° anticlockwise: $(x, y) \to (-y, x)$ — because $\cos 90° = 0$, $\sin 90° = 1$
• 180°: $(x, y) \to (-x, -y)$ — because $\cos 180° = -1$, $\sin 180° = 0$
• 270° anticlockwise (= 90° clockwise): $(x, y) \to (y, -x)$

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[!formula] Enlargement from Center

What it does: Scales shape by factor $k$ from center $(a, b)$

$\text{Image point: } (x', y') = (a, b) + k \cdot [(x, y) - (a, b)]$

Expanded: $x' = a + k(x - a) = kx + a(1-k)$ $y' = b + k(y - b) = ky + b(1-k)$

Why this formula?

1. Find displacement vector from center to point: $(x-a, y-b)$
2. Scale this displacement by $k$: $k(x-a, y-b)$
3. Add back the center: $(a,b) + k(x-a, y-b)$

Intuition:

• If $k=2$, the point moves twice as far from the center
• If $k=0.5$, it moves halfway toward the center
• If $k=1$, point stays put (no enlargement)
• If center is origin $(0,0)$: simply $(x', y') = (kx, ky)$

Special case — Origin as center: $x' = kx, \quad y' = ky$

Why area scales by $k^2$? If you enlarge by factor $k$, every linear dimension multiplies by $k$. Area = length × width, so area multiplies by $k \times k = k^2$.

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Worked Examples

[!example] Example 1: Translation

Problem: Translate triangle with vertices $A(1, 2)$, $B(3, 2)$, $C(2, 4)$ by vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$.

Solution: Apply $(x', y') = (x+2, y-3)$ to each vertex:

• $A(1, 2) \to A'(1+2, 2-3) = A'(3, -1)$
• $B(3, 2) \to B'(3+2, 2-3) = B'(5, -1)$
• $C(2, 4) \to C'(2+2, 4-3) = C'(4, 1)$

Why this step? Each coordinate shifts by the vector components: +2 horizontally (right), -3 vertically (down).

Check: The shape has moved but not changed size or orientation. Side $AB$ length before: $3-1=2$. After: $5-3=2$. ✓

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[!example] Example 2: Reflection in x-axis

Problem: Reflect point $P(4, 5)$ in the $x$-axis.

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

$P(4, 5) \to P'(4, -5)$

Why? The $x$-axis is $y=0$. Point $P$ is 5 units above; $P'$ must be 5 units below. The $x$-coordinate doesn't change because reflection is vertical flip.

Check: Distance from $P$ to axis = $5$. Distance from $P'$ to axis = $5$. ✓

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[!example] Example 3: Rotation 90° anticlockwise around origin

Problem: Rotate point $Q(3, 1)$ by 90° anticlockwise around origin.

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

$Q(3, 1) \to Q'(-1, 3)$

Why this step?

• At 90°: $\cos 90° = 0$, $\sin 90° = 1$
• $x' = 3 \cdot 0 - 1 \cdot 1 = -1$
• $y' = 3 \cdot 1 + 1 \cdot 0 = 3$

Visual check: Original point in1st quadrant (right, up). After 90° anticlockwise rotation, it's in 2nd quadrant (left, up). ✓

Alternative: Distance from origin: $r = \sqrt{3^2+1^2} = \sqrt{10}$. After rotation, distance is still $\sqrt{(-1)^2+3^2} = \sqrt{10}$. ✓

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[!example] Example 4: Enlargement scale factor 2, center (1, 1)

Problem: Enlarge point $R(3, 2)$ with center $(1, 1)$ and scale factor $k=2$.

Solution: Use $x' = 1 + 2(3-1)$, $y' = 1 + 2(2-1)$:

1. Displacement from center: $(3-1, 2-1) = (2, 1)$
2. Scale displacement: $2 \cdot (2, 1) = (4, 2)$
3. Add center: $(1, 1) + (4, 2) = (5, 3)$

Answer: $R'(5, 3)$

Why this step? The point was 2 units right and 1 unit up from center. After doubling, it's 4 units right and 2 units up from center.

Check: Distance from center to $R$: $\sqrt{(3-1)^2+(2-1)^2} = \sqrt{5}$. Distance to $R'$: $\sqrt{(5-1)^2+(3-1)^2} = \sqrt{16+4} = \sqrt{20} = 2\sqrt{5}$. Exactly doubled. ✓

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[!example] Example 5: Combined transformation

Problem: Point $S(2, 3)$ is reflected in the $y$-axis, then translated by $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$. Where does it end up?

Solution:

Step 1 — Reflection in y-axis: $(x, y) \to (-x, y)$ $S(2, 3) \to S_1(-2, 3)$

Why? The $y$-axis is $x=0$. Point is 2 units right; image is 2 units left.

Step 2 — Translation: $(x, y) \to (x+1, y-2)$ $S_1(-2, 3) \to S'(-2+1, 3-2) = S'(-1, 1)$

Why? Add vector components: +1 horizontally, -2 vertically.

Final answer: $S'(-1, 1)$

Important: Order matters! Reflection then translation ≠ translation then reflection.

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Common Mistakes

[!mistake] Mistake 1: Translation vs. Enlargement from origin

Wrong thinking: "Translation and enlargement from origin both move the point, so they're similar."

Why it feels right: Both involve coordinate arithmetic and seem to change position.

The fix:

• Translation: Every point moves by the same fixed vector → shape slides intact
• Enlargement: Points move different amounts depending on distance from center → shape grows/shrinks

For translation by $(2, 3)$: $(1,1) \to (3, 4)$ and $(10, 20) \to (12, 23)$ — both shifted by $(2,3)$.

For enlargement $k=2$ from origin: $(1,1) \to (2, 2)$ and $(10, 20) \to (20, 40)$ — farther points move more.

Steel-man: The confusion arises because both use addition/multiplication. But translation is constant displacement, enlargement is proportional scaling.

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[!mistake] Mistake 2: Reflection across $y=x$ is just $(x, y) \to (y, x)$?

Wrong claim: "Any reflection swaps coordinates."

Why it feels right: The $y=x$ reflection does swap, so students generalize.

The fix: Only $y=x$ and $y=-x$ swap coordinates. Reflections in axes do NOT swap:

• $x$-axis: $(x, y) \to (x, -y)$ — no swap
• $y$-axis: $(x, y) \to (-x, y)$ — no swap

Why $y=x$ is special: It's a diagonal line. Swapping $x$ and $y$ is geometrically equivalent to reflecting across that diagonal. For other lines, the formula is more complex.

Steel-man: Students see the pattern-matching shortcut for $y=x$ and over-apply it.

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[!mistake] Mistake 3: Rotation formula for 90° clockwise

Wrong answer: $(x, y) \to (y, x)$ for90° clockwise.

Why it feels right: Looks similar to reflection in $y=x$.

The fix:

• 90° anticlockwise: $(x, y) \to (-y, x)$
• 90° clockwise: $(x, y) \to (y, -x)$

Check with example: $(2, 0)$ on positive $x$-axis.

• 90° anticlockwise → $(0, 2)$ on positive $y$-axis ✓
• 90° clockwise → $(0, -2)$ on negative $y$-axis ✓

Swapping to $(y, x) = (0, 2)$ is anticlockwise, not clockwise.

Steel-man: The notation looks symmetric, so students confuse the two directions. Remember: clockwise is "opposite" rotation, so one coordinate negates.

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[!mistake] Mistake 4: Enlargement with negative scale factor

Question: What happens if $k = -2$?

Common error: "It just makes the shape smaller."

The fix:

• $k > 1$: enlargement (bigger)
• $0 < k < 1$: reduction (smaller)
• $k = 1$: identity (no change)
• $k < 0$: enlargement and rotation180° around center

For $k=-2$ from origin: $(x, y) \to (-2x, -2y)$ — the shape doubles in size AND flips to opposite side.

Why? The negative sign reverses direction from center, while $|k|=2$ scales the distance.

Steel-man: Students focus on the magnitude and ignore the sign's geometric meaning.

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Active Recall

[!recall]- Explain to a12-year-old

Imagine you're playing with a toy car on a table:

Translation is sliding the car across the table — it moves but doesn't spin or change size. You could write instructions like "move3 squares right and 2 squares forward."

Reflection is putting a mirror next to the car — the mirror car looks identical but reversed, like your left hand becomes a right hand in the mirror.

Rotation is spinning the car around a point — maybe you hold one wheel fixed and turn the car. It faces a different direction but it's still the same car.

Enlargement is like taking a photo of the car and zoming in or out. The car gets bigger or smaller, but the shape stays the same. If you zoom from a point that's not the center of the car, the car also moves.

Each transformation has rules: translation needs a direction and distance, reflection needs a mirror line, rotation needs a center point and angle, enlargement needs a center and a zoom factor (scale factor).

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[!mnemonic] TREE for Transformations

Translation: Teleport with vector (slides)
Reflection: Reverse in mirror (flips)
Rotation: Revolve around center (spins)
Enlargement: Expand from center (scales)

For reflection lines:

• X-axis: "X stays, Y dies" → $(x, -y)$
• Y-axis: "Y stays, X exits" → $(-x, y)$
• Y=X: "Swap" → $(y, x)$

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Connections

• Vectors — translations are vector additions
• Coordinate Geometry — transformations map between coordinates
• Symmetry — reflections and rotations create symmetric patterns
• Similar Triangles — enlargement creates similar shapes
• Matrices — transformations can be represented as matrix operations
• Trigonometry — rotation formulas use sin/cos
• Congruence — translation, reflection, rotation preserve congruence
• Distance Formula — check distances are preserved in rigid transformations
• Inverse Functions — each transformation has an inverse (e.g., translate back)

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#flashcards/maths

What is a translation? :: A transformation that slides every point of a shape by the same fixed vector, preserving shape, size, and orientation.

What is an enlargement? :: A transformation that scales a shape from a center by a scale factor $k$, preserving shape but changing size (unless $k=1$).

What information is needed to describe a rotation? :: The center point, angle, and direction (clockwise/anticlockwise).

What information is needed to describe an enlargement? :: The center point and the scale factor $k$.

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Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Chalo isko simple tarike se samajhte hain. Transformations ka matlab hai ek shape ko move, flip, ghumana, ya uska size badalna. Char basic types hain — translation matlab shape ko slide karna (jaise photo ko table pe khisukana), reflection matlab mirror ki tarah ulta karna, rotation matlab kisi point ke around ghumana, aur enlargement matlab shape ko bada ya chhota karna. Har transformation ke apne rules hain jo batate hain ki kya same rehta hai (jaise angles ya shape) aur kya change hota hai (jaise position ya size). Ye concept isliye important hai kyunki geometry, computer graphics, aur symmetry sab yahi se aate hain.

Ab formulas ka intuition dekho — yeh rattne wali cheez nahi hai, logic se aate hain. Translation mein agar tum vector $\binom{a}{b}$ se move karte ho, toh naya point simply $(x+a, y+b)$ ho jaata hai — matlab har point ko utna hi shift kar do. Reflection ke case mein, jaise x-axis pe flip karo toh $(x,y)$ banjaata hai $(x,-y)$, kyunki x same rehta hai par y ka sign palat jaata hai (upar wala point neeche chala jaata hai, same distance pe). Aur $y=x$ line pe reflect karo toh coordinates swap ho jaate hain, kyunki yeh diagonal line x aur y ka role exchange kar deti hai.

Yeh cheez isliye matter karti hai kyunki aage chalke jab tum vectors aur matrices padhoge, toh yahi transformations base banenge — jaise rotation ka formula ek matrix se aata hai. Agar aaj tum yeh intuition pakad loge ki kaun sa transformation kya preserve karta hai aur kya badalta hai, toh coordinate geometry ke bade sawaal bhi tumhein easy lagenge. So basic idea yaad rakho: har transformation ek instruction hai shape ko move karne ki, aur uska ek clear rule hota hai.