Pythagoras Theorem and its Proof
Hello friends, kaise hain aap log, I hope ki aap log theek hoge.
Friends mera naam hai Dheeraj Sahni aur aap log hai meri website www.mathshindi.com par.Aapka bohot bohot swagat hai meri iss website par. Iss website par aapko maths se related complex topic ko easy way me samjhaaya jaata hai aur sirf ye hi nahi wo complex topics aapko acche se samajh me aa jaaye iske liye aapko unn topics se related examples/questions bhi diye jaate hai.
Friends, aapne apne school time main kabhi Pythagoras Theorem ka naam jaroor suna hoga aur iss theorem ko aapne use bhi kiya hoga, bohot se maths ke problems/questions ko solve karne ke liyeliye. Friends, aaj me iss article mein aapko isi theorem– Pythagoras Theorem (ya Pythagorean Theorem) ke baare mein batane wala hoon aur saath hi saath iss theorem ka prove bhi me aapko doonga. Iss article mein hum Pythagoras Theorem ke converse theorem ke baare mein bhi jaanenge aur iss converse theorem ka proof bhi dekhenge. To chaliye friends start karte hain aaj ka ye article. Iss article ko last tak padhe.
Friends, Pythagoras Theorem ek important aur bohot useful theorem hai. Iss theorem ka use mathematics aur science ke bohot se disciplines mein to hota hi hai aur iska use hamari daily life mein bhi hota hai. To friends, iss theorem– Pythagoras theorem ke baare mein jaanne se pehle, chaliye uss vyakti(person) ke baare mein jaan lete hain jisne iss theorem ko diya tha. Ji haan, main baat kar raha hoon Pythagoras ki.
Biography of Pythagoras
Pythagoras, mathematics ke itihas(history) mein bohot se famous naam(name) mein se ek hai aur Pythagoras ko pehla true mathematician ke roop mein jaana jaata hai. Iss prasiddh(famous/legendary) mathematician se related jo aaj hamare paas jyadatar(most) information hai, wo kai(many) centuries ke baad ki hain jab woh raha karte the. Isliye bahut si information unreliable hain. Unki jo pehle ki biographies hai wo aise authors ke dwara likhi gayi hai jo Pythagoras ko kisi supernatural power ya god like person ki tarah dikhana chahte the. Aisa kaha jaata hai ki Pythagoras ke birth se pehle bhavishyawaani(fortelling/prophesize) kar di gayi thi ki unki pregnant mother ek aise vyakti(person) ko janm(birth) dengi jo bahut hi beautiful, wise aur poore maanavta(humankind) ke liye laabhkaari(beneficial) hoga.
Pythagoras ka janm(birth) Greek Island- Samos par eastern Aegean mein hua tha. Unki birthdate approximately 570 BC bataya jata hai. Unke father Mnesarchus ek vyaapaari(merchant) the aur wo business ke liye bohot travel kiya karte the. Pythagoras bhi apne father ka saath deta tha unke bohot saare expeditions mein. Jab Pythagoras 18 saal(years) ke the to wo Miletus- ek ancient Greek city gaye the joki Anatolia ke western coast mein tha; wahan par wo mile Thales se jo sabse pehle(first) Greek philosopher aur scientist the. Uss time par Thales bohot hi old the aur unhe nahi lagta tha ki Pythagoras ko padhaana(teach karna) koi great deal hai. Phir bhi, Pythagoras ki Thales ke saath hui iss meeting ne Pythagoras ko mathematics, science aur astronomy mein aur jyada interested kar diya. Thales ne unhe Egypt travel karne ki advise di. Aisa believe kiya jaata hai ki Pythagoras ke andar learning ki strong desire thi aur unke isi desire ki wajah se unhone bahut lambe-lambe (lenthy/extensive) travels kiye. wo bahut saare teachers aur philosophers se padhe (teaching li). Unhone, uss waqt Egypt mein jitni bhi available knowledge thi uski khoj(search) mein Egypt me kai saal (many years) bita diye (spent kar diye). Aur unhone ek Egyptian priest Oenuphis, jo Heliopolis naam ki jagah par rehte the, se bhi wisdom prapt(receive) ki thi.
Approximately 530 BC mein Pythagoras Croton mein shift ho gaye jo ki Italy mein tha. Waha par unhone ek philosophical aur religious school khola. Wo school bahut jald(soon) hi famous ho gaya aur bahut saare students/followers bhi waha par aane lage. Unhone ek society ki sthapna(foundation) ki- Mathematickoi. Iss society ko Pythagoras hi chalaya karte the. Iss society members permanently ek saath hi rehte the aur unn members ko strict rules follow karne hote the. Pythagoras uss society ke sabhi(all) members ko personally padhaate the (teach karte the). Unke school/society ke strict rules secrecy aur communal system ki wajah se Pythagoras ka actual work pata nahi chalta. Pythagoras ka actual work aur unke followers ke work ko distinguish karna hard hai.
Geometric ke theorem- Pythagoras Theorem ka credit commonly Pythagoras ko hi diya jaata hai. Halaki(Though) ye theorem pehle se hi Babylonians aur Indians ke dwara use kiya jaata raha hai. Bahut widely aisa believe kiya jaata hai ki Pythagoras ya unke students ne hi sabse pehle iss theorem ka proof diya tha.
Pythagoras hamesha politics se door rehna chahte the, phir bhi unki society hamesha politics se affect hui hai. 510 BC mein Croton ne apne neighbour Sybaris par attack kiya aur usse defeat kar diya aur aise kuch certain suggestions hai jisse ye pata chalta hai ki Pythagoras uss dispute mein involve the. Aur phir approximately 508 BC mein Croton mein jo Pythagorean society thi uss par Cylon ne attack kar diya. Pythagoras waha se Metapontium ko bhaag gaye. Bohot saare authors ye kehte hain ki uss attack mein Pythagoras ki death ho gayi thi. Kuch authors ye bhi kehte hain ki unki society par hue attack ki wajah se unhone suicide kar liya tha. Iss baat ka koi proof/evidence nahi hai ki Pythagoras ki death kab aur kaise hui lekin 500 BC ke baad se unki society rapidly expand hoti gayi aur iss society ke mathematics mein contribution ab bhi recognize kiye jaate hain aur unki respect ki jaati hai.
To ye thi Pythagoras ki short biography.
Chaliye ab hum iss theorem ko dekhte hain jiske use se hum Pythagoras Theorem ko prove karenge.
Theorem- ''If a perpendicular drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other''.
"Agar right triangle mein right angle waale vertex se hypotenuse par perpendicular draw kar diya jaaye to perpendicular ke dono taraf ke triangles uss poore triangle ke aur aapas mein similar honge".
Iss theorem ko mein yaha par proof karke nahi dikhaoonga. Iss theorem ko aap aasaani se khud hi prove kar sakte ho; agar aapko similar triangle ki aur usse related different similarity criterion ke baare mein jaankari hai to.
Agar aap similar triangles aur similar triangles related different similarity criterion ke baare mein jaanna chahte hain to aap mere iss article ko padhiye >> >> >>
Also read:
To chliye ab aate hain main topic par-
Pythagoras theorem (or Pythagorean theorem)
"In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides".
"Kisi right triangle mein, hypotenuse ka square doosri do two sides ke square ke sum ke equal hota hai".
Proof of Pythagoras Theorem
Given: Ek right ∆ABC jisme B par right angle hai.
To prove: AC2 = AB2 + BC2
Construction: Draw kar diya BD ⊥ AC
Proof: Now, ∆ADB~∆ABC (Using above theorem)
So, AD = AB (Sides are
AB AC proportional of similar triangles)
or, AD•AC = AB•AB
or, AD•AC = AB2 ---------(1)
Also, ∆BDC~∆ABC (Using above theorem)
So, CD = BC (Sides are
BC AC proportional of similar triangles)
or, CD•AC = BC•BC
CD•AC = BC2 ---------(2)
Adding (1) and (2),
AD•AC + CD•AC = AB2 + BC2
AC (AD + CD) = AB2 + BC2
AC•AC = AB2 + BC2
AC2 = AB2 + BC2
Yahi Pythagoras Theorem hai.
[Note: Ye theorem, jisko aaj hum Pythagoras Theorem ke naam se jaante hain, ek ancient Indian mathematician Baudhayan ne about 800 B.C.E. mein iss form mein diya tha-
"The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth)".
Yahi reason hai ki kabhi-kabhi Pythagoras theorem ko Baudhayan Theorem bhi kehte hain].
Waise to bahut saare methods hain Pythagoras Theorem ko prove karne ke aur maine yaha par unme se ek method use kiya hai.
Chaliye ab dekhte hain Pythagoras Theorem ke converse theorem ko.
Converse of Pythagoras Theorem
"In a triangle, if square of one side is equal to the sum of the squares of the other two sides then, then the angle opposite the first side is a right angle".
"Kisi triangle mein, agar ek side ka square, doosri dono(two) sides ke square ke sum ke equal ho to pehli(first) side ke saamne(opposite) waala angle right angle hota hai".
Proof of Converse of Pythagoras Theorem
Given: Maan lijiye ek triangle hai ∆ABC jisme AC2 = AB2 + BC2 diya hua hai.
To prove: ∠B = 90°
Construction: Hamne ek doosra triangle ∆PQR draw kar diya jisme Q par right angle hai aur wo triangle iss tarah draw kiya ki PQ = AB aur QR = BC.
Proof: Now, from ∆PQR,
PR2 = PQ2 + QR2 (Pythagoras Theorem as ∠Q = 90°)
or, PR2 = AB2 + BC2 (By construction) --------(1)
But, AC2 = AB2 + BC2 (Given) ---------(2)
So, AC = PR [From (1) and (2)] ---------(3)
Now, in ∆ABC and ∆PQR,
AB = PQ (By construction)
BC = QR (By construction)
AC = PR [Proved in (3) above]
So, ∆ABC ≅ ∆PQR (SSS congruence)
Therefore, ∠B = ∠Q (CPCT)
But ∠Q = 90° (By construction)
So, ∠B = 90°
Chaliye abhi hamne jo kuch bhi seekha hai usse aur achcche se samajhne ke liye inse related kuch solved examples dekh lete hain.
Ex.1. In the given figure, ∠ACB = 90° and CD ⊥ AB. Prove that BC = BD .
AC AD
Solution: ∆ACD~∆ABC (Using the theorem discussed above)
So, AC = AD
AB AC
or, AC2 = AB•AD --------(1)
Similarly, ∆BCD~∆BAD (Using the theorem discussed above)
So, BC = BD
BA BC
or, BC2 = BA•BD --------(2)
Therefore, from (1) and (2),
BC2 =BA•BD
AC2 AB•AD
BC2 = BD
AC2 AD
Ex.2. A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.
Solution: Let AB be the ladder and CA be the wall with the window at A (see the diagram).
BC = 2.5 m and CA = 6 m
From Pythagoras theorem,
AB2 = BC2 + CA2
AB2 = (2.5)2 + (6)2
AB2 = 6.25 + 36
AB2 = 42.25
AB = √42.25
So, AB = 6.5
Thus, length of ladder is 6.5 m.
Ex.3. In the given figure, if AD ⊥ BC, prove that AB2 + CD2 = BD2 + AC2.
Solution: From ∆ADC,
AC2 = AD2 + CD2 (Pythagoras Theorem) -----(1)
From ∆ADB,
AB2 = AD2 + BD2 (Pythagoras Theorem) ------(2)
Subtracting (1) from (2),
AB2 – AC2 = AD2 + BD2 – AD2 – CD2
Or, AB2 – AC2 = BD2 – CD2
Or, AB2 + CD2 = BD2 + AC2
Ex.4. A ladder 10 m long reaches a window 8 m above the ground. Find distance of the foot of the ladder from base of the wall.
Solution: Let the length of ladder be AB, the height of window from the ground be BC and the distance of the foot of ladder from the base of the wall be AC.
So, AB = 10 m and BC = 8 m.
Using Pythagoras Theorem,
AB2 = BC2 + AC2
(10)2 = (8)2 + AC2
100 = 64 + AC2
100 – 64 = AC2
36 = AC2
√36 = AC
6 = AC
So, distance of the foot of the ladder from base of the wall is 6 m.
Ex.5. In a ∆ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm. What is angle B ?
Solution: AB = 6√3 cm
AB = (6√3 cm)2
AB = 108 cm2 --------(1)
and, BC = 6 cm
BC2 = (6 cm)2
BC2 = 36 cm2 -------(2)
Adding (1) and (2),
AB2 + BC2 = 108 cm2 + 36 cm2
AB2 + BC2 = 144 cm2
AB2 + BC2 = (12 cm)2
AB2 + BC2 = AC2
But this is possible in only right triangle. That means, ∆ABC is a right triangle and ∠B = 90°.
I hope ki mera ye article aapko pasand aaya hoga. Agar aapko mera ye article pasand aaya to aap hame comment ke maadhyam se bata sakte hain.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1- ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2?
Q.2- ABC is an equilateral triangle of side 2a. Find each of its altitudes.
Q.3- A ladder 13 m long reaches a window 12 m above the ground. Find distance of the foot of the ladder from base of the wall.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
Friends mera naam hai Dheeraj Sahni aur aap log hai meri website www.mathshindi.com par.Aapka bohot bohot swagat hai meri iss website par. Iss website par aapko maths se related complex topic ko easy way me samjhaaya jaata hai aur sirf ye hi nahi wo complex topics aapko acche se samajh me aa jaaye iske liye aapko unn topics se related examples/questions bhi diye jaate hai.
Friends, aapne apne school time main kabhi Pythagoras Theorem ka naam jaroor suna hoga aur iss theorem ko aapne use bhi kiya hoga, bohot se maths ke problems/questions ko solve karne ke liyeliye. Friends, aaj me iss article mein aapko isi theorem– Pythagoras Theorem (ya Pythagorean Theorem) ke baare mein batane wala hoon aur saath hi saath iss theorem ka prove bhi me aapko doonga. Iss article mein hum Pythagoras Theorem ke converse theorem ke baare mein bhi jaanenge aur iss converse theorem ka proof bhi dekhenge. To chaliye friends start karte hain aaj ka ye article. Iss article ko last tak padhe.
Let's begin...
Friends, Pythagoras Theorem ek important aur bohot useful theorem hai. Iss theorem ka use mathematics aur science ke bohot se disciplines mein to hota hi hai aur iska use hamari daily life mein bhi hota hai. To friends, iss theorem– Pythagoras theorem ke baare mein jaanne se pehle, chaliye uss vyakti(person) ke baare mein jaan lete hain jisne iss theorem ko diya tha. Ji haan, main baat kar raha hoon Pythagoras ki.
Biography of Pythagoras
 |
| Pythagoras |
Pythagoras, mathematics ke itihas(history) mein bohot se famous naam(name) mein se ek hai aur Pythagoras ko pehla true mathematician ke roop mein jaana jaata hai. Iss prasiddh(famous/legendary) mathematician se related jo aaj hamare paas jyadatar(most) information hai, wo kai(many) centuries ke baad ki hain jab woh raha karte the. Isliye bahut si information unreliable hain. Unki jo pehle ki biographies hai wo aise authors ke dwara likhi gayi hai jo Pythagoras ko kisi supernatural power ya god like person ki tarah dikhana chahte the. Aisa kaha jaata hai ki Pythagoras ke birth se pehle bhavishyawaani(fortelling/prophesize) kar di gayi thi ki unki pregnant mother ek aise vyakti(person) ko janm(birth) dengi jo bahut hi beautiful, wise aur poore maanavta(humankind) ke liye laabhkaari(beneficial) hoga.
Pythagoras ka janm(birth) Greek Island- Samos par eastern Aegean mein hua tha. Unki birthdate approximately 570 BC bataya jata hai. Unke father Mnesarchus ek vyaapaari(merchant) the aur wo business ke liye bohot travel kiya karte the. Pythagoras bhi apne father ka saath deta tha unke bohot saare expeditions mein. Jab Pythagoras 18 saal(years) ke the to wo Miletus- ek ancient Greek city gaye the joki Anatolia ke western coast mein tha; wahan par wo mile Thales se jo sabse pehle(first) Greek philosopher aur scientist the. Uss time par Thales bohot hi old the aur unhe nahi lagta tha ki Pythagoras ko padhaana(teach karna) koi great deal hai. Phir bhi, Pythagoras ki Thales ke saath hui iss meeting ne Pythagoras ko mathematics, science aur astronomy mein aur jyada interested kar diya. Thales ne unhe Egypt travel karne ki advise di. Aisa believe kiya jaata hai ki Pythagoras ke andar learning ki strong desire thi aur unke isi desire ki wajah se unhone bahut lambe-lambe (lenthy/extensive) travels kiye. wo bahut saare teachers aur philosophers se padhe (teaching li). Unhone, uss waqt Egypt mein jitni bhi available knowledge thi uski khoj(search) mein Egypt me kai saal (many years) bita diye (spent kar diye). Aur unhone ek Egyptian priest Oenuphis, jo Heliopolis naam ki jagah par rehte the, se bhi wisdom prapt(receive) ki thi.
Approximately 530 BC mein Pythagoras Croton mein shift ho gaye jo ki Italy mein tha. Waha par unhone ek philosophical aur religious school khola. Wo school bahut jald(soon) hi famous ho gaya aur bahut saare students/followers bhi waha par aane lage. Unhone ek society ki sthapna(foundation) ki- Mathematickoi. Iss society ko Pythagoras hi chalaya karte the. Iss society members permanently ek saath hi rehte the aur unn members ko strict rules follow karne hote the. Pythagoras uss society ke sabhi(all) members ko personally padhaate the (teach karte the). Unke school/society ke strict rules secrecy aur communal system ki wajah se Pythagoras ka actual work pata nahi chalta. Pythagoras ka actual work aur unke followers ke work ko distinguish karna hard hai.
Geometric ke theorem- Pythagoras Theorem ka credit commonly Pythagoras ko hi diya jaata hai. Halaki(Though) ye theorem pehle se hi Babylonians aur Indians ke dwara use kiya jaata raha hai. Bahut widely aisa believe kiya jaata hai ki Pythagoras ya unke students ne hi sabse pehle iss theorem ka proof diya tha.
Pythagoras hamesha politics se door rehna chahte the, phir bhi unki society hamesha politics se affect hui hai. 510 BC mein Croton ne apne neighbour Sybaris par attack kiya aur usse defeat kar diya aur aise kuch certain suggestions hai jisse ye pata chalta hai ki Pythagoras uss dispute mein involve the. Aur phir approximately 508 BC mein Croton mein jo Pythagorean society thi uss par Cylon ne attack kar diya. Pythagoras waha se Metapontium ko bhaag gaye. Bohot saare authors ye kehte hain ki uss attack mein Pythagoras ki death ho gayi thi. Kuch authors ye bhi kehte hain ki unki society par hue attack ki wajah se unhone suicide kar liya tha. Iss baat ka koi proof/evidence nahi hai ki Pythagoras ki death kab aur kaise hui lekin 500 BC ke baad se unki society rapidly expand hoti gayi aur iss society ke mathematics mein contribution ab bhi recognize kiye jaate hain aur unki respect ki jaati hai.
To ye thi Pythagoras ki short biography.
Chaliye ab hum iss theorem ko dekhte hain jiske use se hum Pythagoras Theorem ko prove karenge.
Theorem- ''If a perpendicular drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other''.
"Agar right triangle mein right angle waale vertex se hypotenuse par perpendicular draw kar diya jaaye to perpendicular ke dono taraf ke triangles uss poore triangle ke aur aapas mein similar honge".
Iss theorem ko mein yaha par proof karke nahi dikhaoonga. Iss theorem ko aap aasaani se khud hi prove kar sakte ho; agar aapko similar triangle ki aur usse related different similarity criterion ke baare mein jaankari hai to.
Agar aap similar triangles aur similar triangles related different similarity criterion ke baare mein jaanna chahte hain to aap mere iss article ko padhiye >> >> >>
Also read:
To chliye ab aate hain main topic par-
Pythagoras theorem (or Pythagorean theorem)
"In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides".
"Kisi right triangle mein, hypotenuse ka square doosri do two sides ke square ke sum ke equal hota hai".
Proof of Pythagoras Theorem
Given: Ek right ∆ABC jisme B par right angle hai.
To prove: AC2 = AB2 + BC2
Construction: Draw kar diya BD ⊥ AC
Proof: Now, ∆ADB~∆ABC (Using above theorem)
So, AD = AB (Sides are
AB AC proportional of similar triangles)
or, AD•AC = AB•AB
or, AD•AC = AB2 ---------(1)
Also, ∆BDC~∆ABC (Using above theorem)
So, CD = BC (Sides are
BC AC proportional of similar triangles)
or, CD•AC = BC•BC
CD•AC = BC2 ---------(2)
Adding (1) and (2),
AD•AC + CD•AC = AB2 + BC2
AC (AD + CD) = AB2 + BC2
AC•AC = AB2 + BC2
AC2 = AB2 + BC2
Yahi Pythagoras Theorem hai.
[Note: Ye theorem, jisko aaj hum Pythagoras Theorem ke naam se jaante hain, ek ancient Indian mathematician Baudhayan ne about 800 B.C.E. mein iss form mein diya tha-
"The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth)".
Yahi reason hai ki kabhi-kabhi Pythagoras theorem ko Baudhayan Theorem bhi kehte hain].
Waise to bahut saare methods hain Pythagoras Theorem ko prove karne ke aur maine yaha par unme se ek method use kiya hai.
Chaliye ab dekhte hain Pythagoras Theorem ke converse theorem ko.
Converse of Pythagoras Theorem
"In a triangle, if square of one side is equal to the sum of the squares of the other two sides then, then the angle opposite the first side is a right angle".
"Kisi triangle mein, agar ek side ka square, doosri dono(two) sides ke square ke sum ke equal ho to pehli(first) side ke saamne(opposite) waala angle right angle hota hai".
Proof of Converse of Pythagoras Theorem
Given: Maan lijiye ek triangle hai ∆ABC jisme AC2 = AB2 + BC2 diya hua hai.
To prove: ∠B = 90°
Construction: Hamne ek doosra triangle ∆PQR draw kar diya jisme Q par right angle hai aur wo triangle iss tarah draw kiya ki PQ = AB aur QR = BC.
Proof: Now, from ∆PQR,
PR2 = PQ2 + QR2 (Pythagoras Theorem as ∠Q = 90°)
or, PR2 = AB2 + BC2 (By construction) --------(1)
But, AC2 = AB2 + BC2 (Given) ---------(2)
So, AC = PR [From (1) and (2)] ---------(3)
Now, in ∆ABC and ∆PQR,
AB = PQ (By construction)
BC = QR (By construction)
AC = PR [Proved in (3) above]
So, ∆ABC ≅ ∆PQR (SSS congruence)
Therefore, ∠B = ∠Q (CPCT)
But ∠Q = 90° (By construction)
So, ∠B = 90°
Chaliye abhi hamne jo kuch bhi seekha hai usse aur achcche se samajhne ke liye inse related kuch solved examples dekh lete hain.
Ex.1. In the given figure, ∠ACB = 90° and CD ⊥ AB. Prove that BC = BD .
AC AD
Solution: ∆ACD~∆ABC (Using the theorem discussed above)
So, AC = AD
AB AC
or, AC2 = AB•AD --------(1)
Similarly, ∆BCD~∆BAD (Using the theorem discussed above)
So, BC = BD
BA BC
or, BC2 = BA•BD --------(2)
Therefore, from (1) and (2),
BC2 =BA•BD
AC2 AB•AD
BC2 = BD
AC2 AD
Ex.2. A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.
Solution: Let AB be the ladder and CA be the wall with the window at A (see the diagram).
BC = 2.5 m and CA = 6 m
From Pythagoras theorem,
AB2 = BC2 + CA2
AB2 = (2.5)2 + (6)2
AB2 = 6.25 + 36
AB2 = 42.25
AB = √42.25
So, AB = 6.5
Thus, length of ladder is 6.5 m.
Ex.3. In the given figure, if AD ⊥ BC, prove that AB2 + CD2 = BD2 + AC2.
Solution: From ∆ADC,
AC2 = AD2 + CD2 (Pythagoras Theorem) -----(1)
From ∆ADB,
AB2 = AD2 + BD2 (Pythagoras Theorem) ------(2)
Subtracting (1) from (2),
AB2 – AC2 = AD2 + BD2 – AD2 – CD2
Or, AB2 – AC2 = BD2 – CD2
Or, AB2 + CD2 = BD2 + AC2
Ex.4. A ladder 10 m long reaches a window 8 m above the ground. Find distance of the foot of the ladder from base of the wall.
Solution: Let the length of ladder be AB, the height of window from the ground be BC and the distance of the foot of ladder from the base of the wall be AC.
So, AB = 10 m and BC = 8 m.
Using Pythagoras Theorem,
AB2 = BC2 + AC2
(10)2 = (8)2 + AC2
100 = 64 + AC2
100 – 64 = AC2
36 = AC2
√36 = AC
6 = AC
So, distance of the foot of the ladder from base of the wall is 6 m.
Ex.5. In a ∆ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm. What is angle B ?
Solution: AB = 6√3 cm
AB = (6√3 cm)2
AB = 108 cm2 --------(1)
and, BC = 6 cm
BC2 = (6 cm)2
BC2 = 36 cm2 -------(2)
Adding (1) and (2),
AB2 + BC2 = 108 cm2 + 36 cm2
AB2 + BC2 = 144 cm2
AB2 + BC2 = (12 cm)2
AB2 + BC2 = AC2
But this is possible in only right triangle. That means, ∆ABC is a right triangle and ∠B = 90°.
I hope ki mera ye article aapko pasand aaya hoga. Agar aapko mera ye article pasand aaya to aap hame comment ke maadhyam se bata sakte hain.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1- ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2?
Q.2- ABC is an equilateral triangle of side 2a. Find each of its altitudes.
Q.3- A ladder 13 m long reaches a window 12 m above the ground. Find distance of the foot of the ladder from base of the wall.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
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