Class 7 Maths Chapter 7: A Tale of Three Intersecting Lines (Hinglish Notes)

Ye chapter kis baare mein hai?

Triangle sabse saral band aakriti hai: teen vertices, unhe jodti teen sides, aur har kone par ek angle. ∆ABC mein angles ∠CAB, ∠ABC aur ∠BCA hain, jinhe seedhe ∠A, ∠B aur ∠C likhte hain, aur vertices ko kisi bhi kram mein likh sakte hain. Agar teeno points ek hi seedhi line par hon, to triangle banta hi nahi. Chapter poochhta hai ki triangle sahi sahi kaise banaayein, triangle ban bhi sakta hai ya nahi, aur iske angles ke baare mein kya hamesha sach hai.

Teen sides se triangle banaana

4 cm side waala equilateral triangle sirf ruler se banaana ho, to teesra point andaaze se baar baar lagaana padega. Compass ye ek baar mein kar deta hai: base AB = 4 cm kheencho, A se 4 cm radius ka arc aur B se ek aur arc kheencho, aur jahan ve milein us point C ko A aur B se jod do. C dono arcs par hai, isliye dono siron se 4 cm door hai. Yahi tareeka kisi bhi teen sides se triangle banaata hai, jaise 4 cm, 5 cm aur 6 cm: ek side ko base banaao, phir uske siron se baaki do lambaaiyon ke arcs. Padho triangle kaise banayein.

Teen lambaaiyon se triangle kab banta hai?

3 cm, 4 cm aur 8 cm aazmaao: arcs kabhi nahi milte. Wajah seedhi hai. Tent se ped tak seedha jaana khambe se hokar jaane se chhota hai, isliye har triangle mein do vertices ke beech ki seedhi side teesre vertex se ghoomkar jaane waale raaste se chhoti hoti hai. 10, 15 aur 30 ke liye ye toot jaata hai, kyunki 30 to 10 + 15 se zyaada hai, isliye aisa triangle nahi banta. Teen lambaaiyan triangle inequality tab poori karti hain jab har ek baaki do ke jod se kam ho, aur sabse lambi ko jaanchna kaafi hai. Sabse lambi side ke siron se bane do circles dikhaate hain ki ye niyam kaafi kyun hai: do chhoti lambaaiyon ka jod sabse lambi ke baraabar ho to ve bas chhoote hain, kam ho to milte nahi, aur zyaada ho to ek doosre ko kaatte hain aur triangle ban jaata hai. Padho triangle inequality aur angle sum property.

Do sides aur ek angle, ya do angles aur ek side

Do sides aur unke beech ka angle, yaani included angle, diya ho to ek side kheencho, uske sire par angle banaao, aur nayi arm par doosri side naapo; 180° se kam kisi bhi included angle ke liye triangle banta hai. Do angles aur unke beech ki side, yaani included side, di ho to side kheencho aur uske dono siron par angles banaao; arms teesre vertex par milti hain. Ye tabhi hota hai jab dono angles ka jod 180° se kam ho. A par 40° ho, to B se 140° par kheenchi arm pehli arm ke parallel chalti hai aur use kabhi nahi milti, AB kitni bhi lambi ho.

Angle sum property

Do angles tay kar do, jaise 60° aur 70°, to teesra angle base ki har lambaai par wahi aata hai. Use bina naape nikaalne ke liye upar ke vertex se base ke parallel ek line XY kheencho. Ye line dono sides ke saath jo angles banaati hai, ve alternate angles hain, yaani dono base angles ke baraabar, aur upar ke angle ke saath milkar ek seedha angle banaate hain. Isliye har triangle ke teeno angles ka jod 180° hai. Ye chatur vichaar Euclid ki kitaab The Elements mein milta hai, jo lagbhag 300 BCE mein likhi gayi. Kaagaz ke triangle ke kone modkar base par milaane se bhi yahi dikhta hai. Isi se juda, ek side aage badhaakar bana exterior angle saamne ke do andar ke angles ke jod ke baraabar hota hai.

Altitudes

Kisi vertex ki height saamne ki side tak seedhe neeche naapi jaati hai. Vertex se saamne ki side par perpendicular line segment altitude kehlaata hai, aur har triangle mein teen altitudes hote hain. Obtuse triangle mein base ko aage badhaana padta hai aur altitude triangle ke bahar girta hai. Aisa kaagaz ka mod jo base ko khud par rakhe aur upar ke vertex se guzre, altitude banaata hai, aur ruler ke saath set square ise sahi kheenchta hai. Right-angled triangle mein right angle waali sides khud altitudes hoti hain.

Triangles ke types

Sides ke hisaab se triangle equilateral (teen baraabar sides), isosceles (do baraabar sides) ya scalene (saari sides alag) hota hai. Angles ke hisaab se ye acute-angled (teeno angles acute), right-angled (ek right angle) ya obtuse-angled (ek obtuse angle) hota hai. Acute-angled triangle ke teeno angles acute hone chahiye, kyunki har triangle mein pehle se kam se kam do acute angles hote hain. Padho triangles ke types.

Saar mein

Compass triangle banaane ko kuch saaf steps mein badal deta hai; triangle inequality aur 180° ka niyam pehle hi bata dete hain ki triangle ban sakta hai ya nahi; parallel lines angle sum property saabit karti hain; aur altitudes aur types tasveer poori karte hain. Chapter ek dabbe par chalti makdi ke sabse chhote raaste ki paheli se khatam hota hai. Har sawaal ke jawaab Chapter 7 ke NCERT solutions mein hain.