Pythagorean theorem — proof (by similar triangles, rearrangement), converse

2.4.1 · Maths › Trigonometry — Foundation

Overview

Pythagorean theorem Euclidean geometry ki neenv hai, jo kisi bhi right triangle ki teen sides ko aapas mein relate karta hai. Yeh kehta hai ki ek right triangle mein, hypotenuse (right angle ke saamne wali sabse lambi side) ka square, baaki dono sides ke squares ke sum ke barabar hota hai.

Theorem

Yeh kyun important hai: Yeh theorem algebra aur geometry ko bridge karta hai. Yeh humein distances compute karne deta hai, distance formula prove karta hai, trigonometry ki neenv hai, aur vector analysis, complex numbers, aur yahan tak ki special relativity mein bhi dikhta hai.

Proof 1: Similar Triangles (Algebraic)

Setup: Right triangle $A B C$ se shuru karo jisme right angle $C$ par hai. $C$ se hypotenuse $A B$ par ek perpendicular daalo, jo point $D$ par milti hai.

Perpendicular kyun daalna? Yeh teen similar triangles banata hai: original $△ A B C$ aur do chhote triangles $△ A C D$ aur $△ C B D$ . Similar triangles ki sides proportional hoti hain — hum iska faayda uthayenge.

Step-by-step derivation

Chaliye denote karte hain:

Leg $B C = a$ , leg $A C = b$ , hypotenuse $A B = c$
Perpendicular $C D = h$
Segments $A D = x$ aur $D B = y$ , toh $x + y = c$

Step 1: Similar triangles identify karo.

Teeno triangles mein same angles hain:

$△ A B C \sim △ A C D \sim △ C B D$

Yeh similar kyun hain? Har ek mein ek right angle hai, aur yeh acute angles share karte hain. AA similarity criterion.

Step 2: Proportionality relationships likho.

$△ A B C \sim △ A C D$ ke liye: $\frac{A C}{A B} = \frac{A D}{A C}$

Yeh ratio kyun? Hum corresponding sides match kar rahe hain. $A C$ chhote triangle $△ A C D$ ke hypotenuse se correspond karta hai.

Lengths substitute karte hain: $\frac{b}{c} = \frac{x}{b}$

Cross-multiply karo: $b^{2} = c x ..(1)$

Step 3: Doosre pair par similarity apply karo.

$△ A B C \sim △ C B D$ ke liye: $\frac{B C}{A B} = \frac{B D}{B C}$

$\frac{a}{c} = \frac{y}{a}$

Cross-multiply karo: $a^{2} = cy ...(2)$

Step 4: Equations (1) aur (2) ko add karo.

$a^{2} + b^{2} = cy + c x = c (y + x)$

Add kyun karte hain? Humein ek side par $a^{2} + b^{2}$ chahiye. Dhyaan do ki $x + y = c$ (do segments milkar hypotenuse banate hain).

$a^{2} + b^{2} = c \cdot c = c^{2}$

∴ Proved! $a^{2} + b^{2} = c^{2}$

Proof 2: Rearrangement (Geometric)

Badi idea: Ek square ke around ek hi right triangle ki chaar copies arrange karo, phir area ko do alag tareekon se compare karo.

Step 1: Outer square construction.

Legs $a, b$ aur hypotenuse $c$ wale chaar identical right triangles lo. Unhe ek square ki perimeter ke around aise arrange karo ki:

Hypotenuses outer edges banayein
Legs ek tilted inner square banayein

Outer square ki side length $a + b$ hogi (ek leg plus doosri leg).

Step 2: Total area calculate karo — Method 1.

Outer square area: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ $Total area = 4 \cdot \frac{1}{2} ab + c^{2} = 2 ab + c^{2}$ $4 \times \frac{1}{2} ab + c^{2} = 2 ab + c^{2}$

Step 4: Dono expressions ko equate karo.

Kyunki dono same total area represent karte hain: $a^{2} + 2 ab + b^{2} = 2 ab + c^{2}$

Dono sides se $2 ab$ subtract karo: $a^{2} + b^{2} = c^{2}$

∴ Proved! Yeh visual proof dikhata hai ki theorem area conservation ke baare mein hai.

The Converse

Converse important kyun hai? Yeh right angles ka ek test hai. Tum sides measure karke right angle verify kar sakte ho, bina protractor ki zaroorat ke.

Proof of the Converse

Diya gaya: Triangle jisme sides $a, b, c$ hain aur $c^{2} = a^{2} + b^{2}$ hai.

Prove karna hai: $c$ ke opposite angle ek right angle hai.

Strategy: Ek jaana-pehchana right triangle construct karo aur prove karo ki hamaara triangle uske congruent hai.

Step 1: Ek reference right triangle construct karo.

Legs $a$ aur $b$ wala ek right triangle banao. Pythagorean theorem (forward direction) se, uski hypotenuse ki length $a^{2} + b^{2}$ hogi.

Step 2: Hypotenuses compare karo.

Diya hai: $c^{2} = a^{2} + b^{2}$ , toh $c = a^{2} + b^{2}$ .

Reference triangle ki hypotenuse bhi $a^{2} + b^{2}$ ke barabar hai.

Isliye: Dono triangles ki teeno sides equal hain ( $a, b, c$ ).

Step 3: SSS congruence apply karo.

SSS (Side-Side-Side) congruence se, dono triangles congruent hain.

Conclusion: Kyunki reference triangle mein ek right angle hai, aur triangles congruent hain, hamaare original triangle mein bhi ek right angle hona chahiye. ∎

Common Mistakes

Recall Ek 12-saal ke bachche ko explain karo

Socho tumhare paas ek seedi hai jo ek wall se tikki hai. Seedi woh lambi tirchi cheez hai (hum use hypotenuse kehte hain). Wall ek seedhi side hai jo upar jaati hai, aur zameen ek aur seedhi side hai jo aage-peeche jaati hai.

Pythagorean theorem ek magic rule hai jo kehta hai: maap lo ki seedi wall par kitna upar pahunchi, us number ko square karo. Phir maap lo ki seedi ka nichla sira wall se kitna door hai, aur use square karo. Un dono squared numbers ko add karo.

Tumhe exactly seedi ki length ka square milega!

Kyun? Socho ki teeno sides par squares draw kar rahe ho. Seedi wale square ka area dono doosre squares ke area ke barabar hai. Yeh ek perfect puzzle ki tarah hai jahan do chhote area wale squares perfectly bade square mein fit ho jaate hain.

Aur agar teen sticks mili jisme yeh rule kaam kare (jaise 3 cm, 4 cm, 5 cm), toh tum unhe hamesha ek perfect corner mein arrange kar sakte ho — ek right angle!

Key Formulas Summary

Connections

Distance Formula in Coordinate Geometry — Pythagorean theorem ka direct application
Pythagorean Triples — integer solutions jaise 3-4-5, 5-12-13
Trigonometric Ratios — right triangles se derive hoti hain, sine aur cosine ki neenv
Law of Cosines — non-right triangles ka generalization
3D Distance Formula — $d = x^{2} + y^{2} + z^{2}$ tak extend hota hai
Complex Numbers — magnitude $∣ z ∣ = a^{2} + b^{2}$ for $z = a + bi$
Vectors and Magnitude — $∥ v ∥ = v_{x}^{2} + v_{y}^{2}$

#flashcards/maths

Right triangle ke liye Pythagorean theorem kya kehta hai? :: Legs $a$ aur $b$ , aur hypotenuse $c$ wale right triangle ke liye: $c^{2} = a^{2} + b^{2}$

Hypotenuse kya hota hai?

Right triangle ki sabse lambi side, right angle ke opposite.

Similar triangles proof mein, hum right angle se hypotenuse par perpendicular kyun daalaate hain?

Yeh teen similar triangles banata hai jinki proportional sides

b^{2} = c x

aur

a^{2} = cy

tak le jaati hain, jo add hokar

a^{2} + b^{2} = c^{2}

deti hain.

Pythagorean theorem ka converse kya hai?

Agar

a^{2} + b^{2} = c^{2}

hai (jisme

c

sabse lamba hai), toh triangle mein side

c

ke opposite ek right angle hai.

Tum kaise test karoge ki 8, 15, 17 sides wala triangle right triangle hai? :: Check karo $8^{2} + 1 5^{2} = 1 7^{2}$ hai ya nahi. Compute karo: $64 + 225 = 289 = 289$ ✓ Toh haan, yeh right triangle hai.

Agar ek right triangle ki legs 6 aur 8 hain, toh hypotenuse kya hai?

c = 6^{2} + 8^{2} = 36 + 64 = 100 = 10

Agar ek right triangle ka hypotenuse 13 hai aur ek leg 5 hai, toh doosri leg kya hai?

a = 1 3^{2} - 5^{2} = 169 - 25 = 144 = 12

Pythagorean theorem sirf right triangles ke liye kyun kaam karta hai?

Proof 90° angle par nirbhar karta hai jo specific geometric relationships banata hai (similar triangles ya area arrangements). Non-right triangles ko Law of Cosines chahiye.

Rearrangement proof mein, tilted inner square kya represent karta hai?

Inner square ki side length

c

(hypotenuse) hai, toh uska area

c^{2}

hai.

Yaad karne ke liye ek common Pythagorean triple kya hai?

3-4-5 (aur uske multiples: 6-8-10, 9-12-15, etc.)