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

1.2.17 · Maths › Basic Geometry

Overview

Transformations wo operations hain jo shapes ko move, flip, turn, ya resize karti hain, aur saath mein kuch properties preserve karti hain. Ye symmetry, graphics, aur ye samajhne ki foundation hain ki objects space mein ek doosre se kaise related hain.

---

Core Concepts

[!intuition] Transformations kyun Matter Karti Hain

Transformations ko ek shape move karne ki instructions ki tarah socho. Imagine karo tumhare paas ek photo hai: translation use naye position par slide karta hai, reflection use mirror ki tarah flip karta hai, rotation use ek point ke around spin karta hai, aur enlargement use bada ya chota karta hai. Har transformation ke rules hote hain jo decide karte hain kya same rehta hai (jaise angles ya shape) aur kya badalta hai (jaise position ya size).

Transformations samajhne se tum ye kar sakte ho:

• Predict karna ki shapes movement ke baad kahan land karengi
• Patterns aur symmetry pehchanna
• Coordinate geometry navigate karna
• Baad mein vectors aur matrices ke liye intuition build karna

---

[!definition] Char Basic Transformations

1. Translation: Ek shape ke har point ko same distance mein same direction mein slide karna

   • Ek vector $\left(\begin{array}{c}a\\ b\end{array}\right)$ se describe kiya jata hai jahan $a$ = horizontal shift, $b$ = vertical shift
   • Shape, size, aur orientation unchanged

2. Reflection: Ek shape ko ek mirror line ke upar flip karna

   • Har point apne mirror image par map hota hai jo line se equidistant hota hai
   • Shape aur size unchanged, orientation reversed

3. Rotation: Ek shape ko ek fixed center ke around ek angle se turn karna

   • Zaroori hai: center point, angle (°), direction (clockwise/anticlockwise)
   • Shape aur size unchanged, position orientation change

4. Enlargement: Ek shape ko ek center se ek scale factor se scale karna

   • Scale factor $k>1$ → bada; $0<k<1$ → chota; $k=1$ → same size
   • Shape unchanged, size changes (jab tak $k=1$ na ho), position changes (jab tak center shape par na ho)

---

Derivations & Formulas

[!formula] Translation Vector

Ye kya karta hai: Point $(x,y)$ ko shifts add karke move karta hai

Agar hum vector $\left(\begin{array}{c}a\\ b\end{array}\right)$ se translate karein:

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

Ye formula kyun hai?

• Horizontal shift: $a$ units right (positive) ya left (negative)
• Vertical shift: $b$ units up (positive) ya down (negative)
• Har point same amount move karta hai → shape ek rigid unit ki tarah slide hoti hai

Scratch se Derivation: Translation ka matlab hai "har point same displacement se move karta hai." Agar displacement horizontally $a$ aur vertically $b$ hai, to:

• Original point: $(x,y)$
• Horizontal displacement add karo: $x\to x+a$
• Vertical displacement add karo: $y\to y+b$
• Result: $(x+a,y+b)$

---

[!formula] x-axis ke Paas Reflection

Ye kya karta hai: Points ko $x$-axis (horizontal line $y=0$) ke upar flip karta hai

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

Ye kyun kaam karta hai?

• $x$-coordinate same rehta hai (koi horizontal movement nahi)
• $y$-coordinate sign flip ho jata hai (point axis ke opposite side par move ho jata hai)
• Axis se distance preserved hoti hai: agar point $d$ units upar hai, to image $d$ units neeche hogi

Derivation: $x$-axis line $y=0$ hai. $(x,y)$ ko reflect karne ke liye:

1. $(x,y)$ se axis ki perpendicular distance $|y|$ hai
2. Reflected point opposite side par same distance par hona chahiye
3. Agar original $y>0$ (axis ke upar), to reflected $y^{\prime }<0$ (axis ke neeche) aur $|y^{\prime }|=|y|$
4. Isliye $y^{\prime }=-y$, aur $x^{\prime }=x$ (koi horizontal shift nahi)

Doosre common reflections:

• y-axis $(x=0)$: $(x,y)\to (-x,y)$
• Line $y=x$: $(x,y)\to (y,x)$ — coordinates swap ho jate hain
• Line $y=-x$: $(x,y)\to (-y,-x)$ — swap karo aur negate karo

$y=x$ coordinates kyun swap karta hai? Line $y=x$ diagonal hai. Ek point $(3,1)$ line se 2 units neeche hai; uska reflection $(1,3)$ line se 2 units upar hai. Geometrically, $y=x$ ke across reflect karne se $x$ aur $y$ ke roles exchange ho jaate hain.

---

[!formula] Origin ke Around Rotation

Ye kya karta hai: Point $(x,y)$ ko $(0,0)$ ke around angle $\theta$ anticlockwise spin karta hai

$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

Expand karne par: $x^{\prime }=x\cos \theta -y\sin \theta$ $y^{\prime }=x\sin \theta +y\cos \theta$

Ye formula kyun hai?

$(x,y)$ ko polar coordinates mein socho: origin se distance $r=\sqrt{x^2+y^2}$, positive $x$-axis se angle $\varphi$.

1. Original: $x=r\cos \varphi$, $y=r\sin \varphi$
2. $\theta$ se rotate karne ke baad, naya angle $\varphi +\theta$ hai
3. Naye coordinates: $x^{\prime }=r\cos (\varphi +\theta )=r(\cos \varphi \cos \theta -\sin \varphi \sin \theta )=x\cos \theta -y\sin \theta$ $y^{\prime }=r\sin (\varphi +\theta )=r(\sin \varphi \cos \theta +\cos \varphi \sin \theta )=y\cos \theta +x\sin \theta$

Common angles:

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

---

[!formula] Center se Enlargement

Ye kya karta hai: Shape ko center $(a,b)$ se factor $k$ se scale karta hai

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

Expand karne par: $x^{\prime }=a+k(x-a)=kx+a(1-k)$ $y^{\prime }=b+k(y-b)=ky+b(1-k)$

Ye formula kyun hai?

1. Center se point tak displacement vector nikalo: $(x-a,y-b)$
2. Is displacement ko $k$ se scale karo: $k(x-a,y-b)$
3. Center wapas add karo: $(a,b)+k(x-a,y-b)$

Intuition:

• Agar $k=2$ hai, to point center se double distance par move karta hai
• Agar $k=0.5$ hai, to ye center ki taraf halfway move karta hai
• Agar $k=1$ hai, to point wahi rehta hai (koi enlargement nahi)
• Agar center origin $(0,0)$ hai: simply $(x^{\prime },y^{\prime })=(kx,ky)$

Special case — Origin as center: $x^{\prime }=kx,y^{\prime }=ky$

Area $k^2$ se kyun scale hota hai? Agar tum factor $k$ se enlarge karo, to har linear dimension $k$ se multiply ho jata hai. Area = length × width, isliye area $k\times k=k^2$ se multiply hoti hai.

---

Worked Examples

[!example] Example 1: Translation

Problem: Triangle jiske vertices $A(1,2)$, $B(3,2)$, $C(2,4)$ hain, use vector $\left(\begin{array}{c}2\\ -3\end{array}\right)$ se translate karo.

Solution: Har vertex par $(x^{\prime },y^{\prime })=(x+2,y-3)$ apply karo:

• $A(1,2)\to A^{\prime }(1+2,2-3)=A^{\prime }(3,-1)$
• $B(3,2)\to B^{\prime }(3+2,2-3)=B^{\prime }(5,-1)$
• $C(2,4)\to C^{\prime }(2+2,4-3)=C^{\prime }(4,1)$

Ye step kyun? Har coordinate vector components se shift hota hai: +2 horizontally (right), -3 vertically (down).

Check: Shape move ho gayi hai lekin size ya orientation change nahi hui. Side $AB$ length pehle: $3-1=2$. Baad mein: $5-3=2$. ✓

---

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

Problem: Point $P(4,5)$ ko $x$-axis mein reflect karo.

Solution: $(x,y)\to (x,-y)$ use karo:

$P(4,5)\to P^{\prime }(4,-5)$

Kyun? $x$-axis $y=0$ hai. Point $P$ 5 units upar hai; $P^{\prime }$ 5 units neeche hona chahiye. $x$-coordinate change nahi hota kyunki reflection vertical flip hai.

Check: $P$ se axis ki distance = $5$. $P^{\prime }$ se axis ki distance = $5$. ✓

---

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

Problem: Point $Q(3,1)$ ko origin ke around 90° anticlockwise rotate karo.

Solution: $(x,y)\to (-y,x)$ use karo:

$Q(3,1)\to Q^{\prime }(-1,3)$

Ye step kyun?

• 90° par: $\cos 90°=0$, $\sin 90°=1$
• $x^{\prime }=3\cdot 0-1\cdot 1=-1$
• $y^{\prime }=3\cdot 1+1\cdot 0=3$

Visual check: Original point 1st quadrant mein (right, up). 90° anticlockwise rotation ke baad, ye 2nd quadrant mein hai (left, up). ✓

Alternative: Origin se distance: $r=\sqrt{3^2+1^2}=\sqrt{10}$. Rotation ke baad, distance abhi bhi $\sqrt{(-1{)}^2+3^2}=\sqrt{10}$ hai. ✓

---

[!example] Example 4: Scale factor 2, center (1, 1) ke saath Enlargement

Problem: Point $R(3,2)$ ko center $(1,1)$ aur scale factor $k=2$ ke saath enlarge karo.

Solution: $x^{\prime }=1+2(3-1)$, $y^{\prime }=1+2(2-1)$ use karo:

1. Center se displacement: $(3-1,2-1)=(2,1)$
2. Displacement scale karo: $2\cdot (2,1)=(4,2)$
3. Center add karo: $(1,1)+(4,2)=(5,3)$

Answer: $R^{\prime }(5,3)$

Ye step kyun? Point center se 2 units right aur 1 unit up tha. Double karne ke baad, ye center se 4 units right aur 2 units up hai.

Check: Center se $R$ ki distance: $\sqrt{(3-1{)}^2+(2-1{)}^2}=\sqrt{5}$. $R^{\prime }$ ki distance: $\sqrt{(5-1{)}^2+(3-1{)}^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$. Exactly double ho gayi. ✓

---

[!example] Example 5: Combined transformation

Problem: Point $S(2,3)$ ko $y$-axis mein reflect kiya jaata hai, phir $\left(\begin{array}{c}1\\ -2\end{array}\right)$ se translate kiya jaata hai. Ye kahan end up karta hai?

Solution:

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

Kyun? $y$-axis $x=0$ hai. Point 2 units right hai; image 2 units left hai.

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

Kyun? Vector components add karo: +1 horizontally, -2 vertically.

Final answer: $S^{\prime }(-1,1)$

Important: Order matter karta hai! Reflection phir translation ≠ translation phir reflection.

---

Common Mistakes

[!mistake] Mistake 1: Translation vs. Origin se Enlargement

Galat soch: "Translation aur origin se enlargement dono point ko move karte hain, isliye ye similar hain."

Ye sahi kyun lagta hai: Dono mein coordinate arithmetic involve hoti hai aur dono position change karte lagte hain.

Fix:

• Translation: Har point same fixed vector se move karta hai → shape intact slide hoti hai
• Enlargement: Points alag amounts move karte hain center se distance ke hisaab se → shape grow/shrink hoti hai

$(2,3)$ se translation ke liye: $(1,1)\to (3,4)$ aur $(10,20)\to (12,23)$ — dono $(2,3)$ se shift hue.

Origin se $k=2$ enlargement ke liye: $(1,1)\to (2,2)$ aur $(10,20)\to (20,40)$ — door wale points zyada move karte hain.

Steel-man: Confusion isliye hoti hai kyunki dono addition/multiplication use karte hain. Lekin translation constant displacement hai, enlargement proportional scaling hai.

---

[!mistake] Mistake 2: $y=x$ ke across Reflection sirf $(x,y)\to (y,x)$ hai?

Galat claim: "Koi bhi reflection coordinates swap karta hai."

Ye sahi kyun lagta hai: $y=x$ reflection swap karta hai, isliye students generalize kar lete hain.

Fix: Sirf $y=x$ aur $y=-x$ coordinates swap karte hain. Axes mein reflections swap NAHI karte:

• $x$-axis: $(x,y)\to (x,-y)$ — koi swap nahi
• $y$-axis: $(x,y)\to (-x,y)$ — koi swap nahi

$y=x$ special kyun hai: Ye ek diagonal line hai. $x$ aur $y$ swap karna geometrically us diagonal ke across reflect karne ke barabar hai. Doosri lines ke liye, formula zyada complex hota hai.

Steel-man: Students $y=x$ ke liye pattern-matching shortcut dekhte hain aur use over-apply kar dete hain.

---

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

Galat answer: 90° clockwise ke liye $(x,y)\to (y,x)$.

Ye sahi kyun lagta hai: $y=x$ mein reflection se milta-julta lagta hai.

Fix:

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

Example se check: $(2,0)$ positive $x$-axis par.

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

$(y,x)=(0,2)$ par swap karna anticlockwise hai, clockwise nahi.

Steel-man: Notation symmetric lagta hai, isliye students dono directions confuse kar lete hain. Yaad rakho: clockwise "opposite" rotation hai, isliye ek coordinate negate hoti hai.

---

[!mistake] Mistake 4: Negative scale factor ke saath Enlargement

Question: Kya hoga agar $k=-2$ ho?

Common error: "Ye sirf shape ko chota karta hai."

Fix:

• $k>1$: enlargement (bada)
• $0<k<1$: reduction (chota)
• $k=1$: identity (koi change nahi)
• $k<0$: enlargement aur center ke around 180° rotation

Origin se $k=-2$ ke liye: $(x,y)\to (-2x,-2y)$ — shape double size ho jaati hai AUR opposite side par flip ho jaati hai.

Kyun? Negative sign center se direction reverse kar deta hai, jabki $|k|=2$ distance scale karta hai.

Steel-man: Students magnitude par focus karte hain aur sign ke geometric meaning ko ignore kar dete hain.

---

Active Recall

[!recall]- Ek 12-saal ke bachche ko samjhao

Imagine karo tum ek table par ek toy car ke saath khel rahe ho:

Translation car ko table ke across slide karna hai — ye move karta hai lekin spin ya size change nahi hoti. Tum instructions likh sakte ho jaise "3 squares right aur 2 squares aage move karo."

Reflection car ke paas ek mirror rakhna hai — mirror car bilkul identical lagti hai lekin reversed, jaise mirror mein tumhara left hand right hand ban jaata hai.

Rotation car ko ek point ke around spin karna hai — shayad tum ek wheel fix pakdo aur car ghuma do. Ye alag direction face karta hai lekin abhi bhi same car hai.

Enlargement car ki photo lena aur zoom in ya out karna jaisa hai. Car badi ya choti ho jaati hai, lekin shape same rehti hai. Agar tum ek aise point se zoom karo jo car ka center nahi hai, to car bhi move ho jaati hai.

Har transformation ke rules hain: translation ko ek direction aur distance chahiye, reflection ko ek mirror line chahiye, rotation ko ek center point aur angle chahiye, enlargement ko ek center aur ek zoom factor (scale factor) chahiye.

---

[!mnemonic] Transformations ke liye TREE

Translation: Vector ke saath Teleport (slides) Reflection: Mirror mein Reverse (flips) Rotation: Center ke around Revolve (spins) Enlargement: Center se Expand (scales)

Reflection lines ke liye:

• X-axis: "X rehta hai, Y disappear hota hai" → $(x,-y)$
• Y-axis: "Y rehta hai, X exit karta hai" → $(-x,y)$
• Y=X: "Swap" → $(y,x)$

---

Connections

• Vectors — translations vector additions hain
• Coordinate Geometry — transformations coordinates ke beech map karti hain
• Symmetry — reflections aur rotations symmetric patterns create karte hain
• Similar Triangles — enlargement similar shapes create karta hai
• Matrices — transformations ko matrix operations ki tarah represent kiya ja sakta hai
• Trigonometry — rotation formulas sin/cos use karte hain
• Congruence — translation, reflection, rotation congruence preserve karte hain
• Distance Formula — check karo ki distances rigid transformations mein preserved hain
• Inverse Functions — har transformation ka ek inverse hota hai (jaise wapas translate karna)

---

#flashcards/maths

What is a translation? :: Ek transformation jo shape ke har point ko same fixed vector se slide karta hai, shape, size, aur orientation preserve karta hai.

What is an enlargement? :: Ek transformation jo shape ko ek center se scale factor $k$ se scale karta hai, shape preserve karta hai lekin size change karta hai (jab tak $k=1$ na ho).

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

What information is needed to describe an enlargement? :: Center point aur scale factor $k$.

---

Concept Map