Hello friend !! Aaj hum aapko triangle ka ek important theorem ko proof karna sikhayenge. Aaj hum aapko sikhayenge ki, Kaise proof kare ki, Triangle ke medians ek-dusare ko 2:1 ke ratio me divide karte hai. Iss proof ko start karne se pahle hum aapko suggest karenge ki aap sabse pahle triangle ke baare me achhe se jaan lijiye. Hume maalum hai ki aap triangle ke baare me jaante hai par iss article me aap triangle ke baare me aur adhik knowledge paa sakte hai. Isliye iss ko jarur read kare.
I hope ki ab aap triangle ke baare me acche se jaan chuke honge tab ab hum aapko triangle ke medians ke baare me batate hai.
Median of a triangle-: Sabse pahle hum aapko batade ki median ko hindi me maadhyika (माध्यिका ) kahte hai. Ab hum aapko isski defination batate hai.
Niche diye figure-1 dekhiye.
Uppar diye figure-1. me humne ek equilateral ∆ABC liya hai. (Aap koi bhi triangle le sakte hai) Jisme vertex A aur iss vertex A ke opposite side BC ke mid point D ko milane waala line segment AD ek medians hai.
Issi prakar se BE aur CF bhi iss triangle ke other medians hai.
(1). Hum jaante hai ki, point D line BC ka mid point hai isliye BD = DC . Issi prakar se AE = EC and AF = FB.
(2). Every Maadhyika triangle ko 2 equal triangle me divide karti hai. Inn equal triangle ka area bhi equal hota hai.
For example-: Figure-1 me Maadhyika AD, ∆ABC ko 2 equal triangle ∆ABD aur ∆ADC me divide karti hai.
Issi prakar se Maadhyika BE, 2 equal triangle ∆ABE aur ∆EBC me aur Maadhyika CF, ∆FCB and FCA me divide karti hai.
(3). 3 maadhyika ek-dusre ko ek hi point G par cut karti hai. Ab aapke mind me ye question aa raha hoga ki iss point G ko kya kahte hai ?? Aayiye isske baare me bhi jaan lete hai.
" triangle me wo point jaaha par 3 maadhyika ek - dusre ko ek hi point par cut karti hai uss point ko triangle ka centroid kahte hai."
(4). Centroid , triangle ke medians ko 2:1 ke ratio me divide karta hai.
Or
Triangle ke medians ek-dusre ko 2:1 ke ratio me divide karte hai.
To ye tha friend iss ki property . Hum property 4 ko aaj proof karna sikhenge.
I hope ki ab aap triangle ke baare me acche se jaan chuke honge tab ab hum aapko triangle ke medians ke baare me batate hai.
Median of a triangle-: Sabse pahle hum aapko batade ki median ko hindi me maadhyika (माध्यिका ) kahte hai. Ab hum aapko isski defination batate hai.
" Kisi triangle ke vertex (shirsh) aur uss vertex ke opposite side ke mid point ko milane waale line segment ko medians kahte hai."
Hum jaante hai ki, ek triangle me 3 vertices hote hai to issi aadhar par ek triangle me 3 medians hote hai.Niche diye figure-1 dekhiye.
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| Figure-1 |
Issi prakar se BE aur CF bhi iss triangle ke other medians hai.
Properties of medians of a triangle
Maadhyika ke kuch important property hai jo iss prakar se hai:(1). Hum jaante hai ki, point D line BC ka mid point hai isliye BD = DC . Issi prakar se AE = EC and AF = FB.
(2). Every Maadhyika triangle ko 2 equal triangle me divide karti hai. Inn equal triangle ka area bhi equal hota hai.
For example-: Figure-1 me Maadhyika AD, ∆ABC ko 2 equal triangle ∆ABD aur ∆ADC me divide karti hai.
Issi prakar se Maadhyika BE, 2 equal triangle ∆ABE aur ∆EBC me aur Maadhyika CF, ∆FCB and FCA me divide karti hai.
(3). 3 maadhyika ek-dusre ko ek hi point G par cut karti hai. Ab aapke mind me ye question aa raha hoga ki iss point G ko kya kahte hai ?? Aayiye isske baare me bhi jaan lete hai.
" triangle me wo point jaaha par 3 maadhyika ek - dusre ko ek hi point par cut karti hai uss point ko triangle ka centroid kahte hai."
(4). Centroid , triangle ke medians ko 2:1 ke ratio me divide karta hai.
Or
Triangle ke medians ek-dusre ko 2:1 ke ratio me divide karte hai.
To ye tha friend iss ki property . Hum property 4 ko aaj proof karna sikhenge.
Prove that medians of a triangle divides each other in the ratio 2:1
Chaliye issko proof karna start karte hai. Hum aapko step by step iss ko proof karna sikhayenge.
• Diya hai : Ek ∆ABC jiski medians AD, BE aur CF ek-dusare ko G par cut karti hai. (See figure-2)
• Proof karna hai : AG : GD = BG : GE = CG : GF = 2 : 1
• Construction : medians AD ko point H tak hum itana badayenge (increase karenge) ki, AG = GH ho jaaye. Ab aap BH aur CH ko milaye.
• Proof : Ab hum iska proof start karte hai. Sabse pahle aap ye yaad rakhe ki, kisi triangle ke kisi bhi 2 sides ke mid points ko milane waala line segment uss triangle ki 3rd side ke parallel hota hai. Hum issi rule ka use karne waale hai.
∆ACH me, Side AC ka mid point E aur AH ka mid point G hai.
Isliye EG aur CH ek-dusare ke parallel honge.
Means Hum kah sakte hai ki, GB aur CH ek-dusare ke parallel hai. .......(i)
Ab ∆ABH me, Side AB ka mid point F aur side AH ka mid point G hai. Isliye FG aur BH ek dusare ke parallel hai.
Means, GC aur BH ek dusare ke parallel hai. .......(ii)
Statement (i) aur (ii) se, hum kah sakte hai ki BHCG ek parallelogram (samantar chaturbhuj) hai.
Samantar chaturbhuj BHCG ke diagonal (vikarn) BC aur GH ek dusare ko point D par cut karte hai. Isliye GD = DH = 1/2 GH
Lekin humane jo construction kiya hai usske hisaab se GH = AG
Isliye GD = 1/2 AG
⇒ AG = 2
GD 1
⇒ AG : GD = 2:1
Issi prakar se hum proof kar sakte hai ki
BG : GE = 2 : 1 aur CG : GF = 2 : 1
So AG : GD = BG : GE = CG : GF = 2:1
Friend , I hope ki, aapko aaj ka ye proof achhe se samjh aaya hoga. Phir bhi agar kisi bhi jagah par aapko koi bhi problem ya doubt ho to aap humare facebook page par massage ya niche diye comment box me comment karke puchh sakte hai.Hum aapki problem ya doubt ko dur karne ki puri koshish karenge.
Friend agar aapko lagta hai ki, iss article se kisi other ki thodi bhi help ho sakti hai to please issko share karke humari help kare.
Thanks for reading !!
Also read-:
• Fast multiply by 11 of any numbers in hindi
• Fast calculate square of numbers ending with 5
• Diya hai : Ek ∆ABC jiski medians AD, BE aur CF ek-dusare ko G par cut karti hai. (See figure-2)
• Proof karna hai : AG : GD = BG : GE = CG : GF = 2 : 1
• Construction : medians AD ko point H tak hum itana badayenge (increase karenge) ki, AG = GH ho jaaye. Ab aap BH aur CH ko milaye.
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| Figure-2 |
• Proof : Ab hum iska proof start karte hai. Sabse pahle aap ye yaad rakhe ki, kisi triangle ke kisi bhi 2 sides ke mid points ko milane waala line segment uss triangle ki 3rd side ke parallel hota hai. Hum issi rule ka use karne waale hai.
∆ACH me, Side AC ka mid point E aur AH ka mid point G hai.
Isliye EG aur CH ek-dusare ke parallel honge.
Means Hum kah sakte hai ki, GB aur CH ek-dusare ke parallel hai. .......(i)
Ab ∆ABH me, Side AB ka mid point F aur side AH ka mid point G hai. Isliye FG aur BH ek dusare ke parallel hai.
Means, GC aur BH ek dusare ke parallel hai. .......(ii)
Statement (i) aur (ii) se, hum kah sakte hai ki BHCG ek parallelogram (samantar chaturbhuj) hai.
Samantar chaturbhuj BHCG ke diagonal (vikarn) BC aur GH ek dusare ko point D par cut karte hai. Isliye GD = DH = 1/2 GH
Lekin humane jo construction kiya hai usske hisaab se GH = AG
Isliye GD = 1/2 AG
⇒ AG = 2
GD 1
⇒ AG : GD = 2:1
Issi prakar se hum proof kar sakte hai ki
BG : GE = 2 : 1 aur CG : GF = 2 : 1
So AG : GD = BG : GE = CG : GF = 2:1
Friend , I hope ki, aapko aaj ka ye proof achhe se samjh aaya hoga. Phir bhi agar kisi bhi jagah par aapko koi bhi problem ya doubt ho to aap humare facebook page par massage ya niche diye comment box me comment karke puchh sakte hai.Hum aapki problem ya doubt ko dur karne ki puri koshish karenge.
Friend agar aapko lagta hai ki, iss article se kisi other ki thodi bhi help ho sakti hai to please issko share karke humari help kare.
Thanks for reading !!
Also read-:
• Fast multiply by 11 of any numbers in hindi
• Fast calculate square of numbers ending with 5
Hi apko ittne maths ke questions ate hain apne kaise sikha ya kisi or se sikhte ho
जवाब देंहटाएंAap bhi sikh sakte hai. Hum aapko sikhayenge.
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