Pythagoras Theorem
\color{blue}{a}^2 + \color{green}{b}^2 = \color{orange}{c}^2Transcript
Doston, aaj hum ek aisi theorem seekhenge jo triangles ki duniya mein sabse famous hai! Pehle ek right triangle draw karte hain, jisme ek angle exactly 90 degree hota hai. Is triangle ki teen sides hain, inhe dhyan se dekho aur yaad karo.
Yeh blue wali side hai leg a, jo vertically khadi hai. Yeh green wali side hai leg b, jo horizontally leti hai. Aur yeh orange wali side c hai, jise hum hypotenuse kehte hain, jo right angle ke saamne hoti hai.
- Hypotenuse = right angle ke saamne
Ab Pythagoras ne kuch magical observe kiya, har side par ek square banao! Hypotenuse par bhi ek square banate hain, aur dekho kya hota hai. Pythagoras ne discover kiya ki do chhote squares ka area milake bade square ke area ke barabar hota hai!
\color{blue}{a}^2 + \color{green}{b}^2 = \color{orange}{c}^2Ab ek simple example lete hain, a equals 3 aur b equals 4 hai, c nikalna hai.
3^2 + 4^2 = c^2Pehle 3 squared calculate karte hain, jo hoga 9. Ab 4 squared calculate karte hain, jo hoga 16. Dono add karo, 9 plus 16 equals 25, toh c squared equals 25.
9 + 16 = c^2C squared 25 hai, toh c equals square root of 25, jo exactly 5 hai!
- 3-4-5 ek famous Pythagorean triple hai!
c = \sqrt{25} = 5Ab visual proof dekhte hain, yeh squares actually kaise fit hote hain ek doosre mein. Dekho, yeh tilted square c squared hai, ab hum triangles ko rearrange karte hain proof ke liye. Waah! Triangles move hone ke baad a squared aur b squared ke holes exactly fit ho gaye, yahi proof hai!
Ek common galti, kabhi bhi kisi bhi side ko hypotenuse mat samajh lena, hypotenuse hamesha right angle ke saamne hoti hai. Real life mein, ek ladder wall ke saath lean karti hai, hum uski length Pythagoras se nikaal sakte hain. Wall 6 meter, base 8 meter, toh ladder ki length hogi square root of 36 plus 64, jo 10 meter hai.
- Galti: Hypotenuse = sabse badi side, right angle ke saamne
\sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10 \text{ m}Ab tumhari baari, ek triangle mein a equals 5, b equals 12 hai, toh c kitna hoga? Toh yaad rakho, right triangle mein a squared plus b squared equals c squared, yahi hai Pythagoras theorem!
- a² + b² = c² hamesha right triangle mein!
Quick quiz
Q1. Agar a = 5 aur b = 12 hai, toh hypotenuse c kitni hogi?
5 squared plus 12 squared equals 25 plus 144 equals 169, aur square root of 169 equals 13.
