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Proof On Pythagorean Theorem




In this section you will learn the Proof On Pythagorean Theorem.

Theorem 1 : In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Proof On Pythagorean Theorem
Given : A right angled triangle ABC in which ∠B = 90 0

Prove that : AC 2 = AB 2 + BC 2

Construction : From B draw BD ⊥ AC.


Statements
Reasons
1) ∠ABC = 900 1) Given
2) BD ⊥ AC 2) By construction
3) ∠ADB = 900 3) By definition of perpendicular
4) ∠ADB = ∠ABC 4) Each of 90900
5) ∠A = ∠A 5) Reflexive
6) ΔADB ~ ΔABC 6) By AA postulate
7) AD/AB = AB/AC 7) By basic proportionality theorem
8) AB2 = AD x AC 8) By cross multiplication
9) ∠CDB = ∠ABC 9) Each 900
10) ∠C = ∠C 10) Reflexive
11) ΔBDC ~ ΔABC 11) By AA postulate
12) DC/BC = BC/AC 12) By basic proportionality theorem
13) BC2 = DC x AC 13) By cross multiplication
14) AB2 + BC2 = AD x AC + AC x DC 14) By adding (8) and (13)
15) AB2 + BC2 = AC(AD + DC) 15) By distributive property
16) AB2 + BC2= AC2 16) As AC = AD + DC and by substitution

Application based on Proof On Pythagorean Theorem.
Theorem 2 : ΔABC is an obtuse triangle, obtuse angled at B. If AD ⊥ CB, prove that AC2 = AB2 + BC2 + 2 BC x BD.

Given : ΔABC is an obtuse triangle, obtuse angled at B. AD ⊥ CB.

Prove that : AC 2 = AB 2 + BC 2 + 2 BC x BD

Proof :


In ΔADB,
AB
2 = AD 2 + DB 2 ( Pythagorean theorem)

In ΔADC,
AC
2 = AD 2 + DC 2

AC
2 = AD 2 + (DB + BC) 2 ( since DC = DB + BC)

AC
2 = AD 2 + DB 2 +BC 2 + 2BD x BC [ (a +b) 2 = a 2 + b 2 + 2ab ---> identity]

AC
2 = (AD 2 + DB 2 ) + BC 2 + 2BC x BD

AC
2 = AB 2 + BC 2 + 2BC x BD [ from above ]

This theorem is known as Apollonius theorem.

Statements of some useful theorems

Theorem 1 : Prove that in any triangle, the sum of the squares of any two sides is equal to the twice the square of half of the third side together with twice the square of the median which bisects the third side.


AB
2 + AC 2 = 2 ( AD 2 + BD 2 )

Theorem 2 : ∠B of ΔABC is an acute angle and AD ⊥ BC.



AC
2 = AB 2 + BC 2 - 2 BC x BD
Pythagorean Theorem

Introduction of Pythagorean Theorem
Converse of Pythagorean Theorem
Pythagorean Triples
Application On Pythagorean Theorem
Proof on Pythagorean Theorem

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