Pythagoras Theorem


 
 
Concept Explanation
 

Pythagoras Theorem

Pythagoras Theorem:

Pythagoras, the famous Greek philosopher, lived about 572 B.C. to 501 B.C. He proved the relation between the lengths of the sides of a triangle, Although, this theorem was known to the Babylonians  1000 years ealier, but, Pythagoras is believed to have the first to discovered a proof of this theorem. However, long ago ( 800 B.C.), the Indian mathe,atician Baudhaya had stated and proved this property of a right angled triangle.

Let us have a look at the fig. large Delta ABC  is a right triangle, right angled at C, so that AB is the hypotenuse and AC and BC are the sides of the right triangle,

Then,                    large (AB)^{2}=(BC)^{2}+(CA)^{2}

i.e.,                 large (Hypotenuse)^{2}=(Base)^{2}+(perpendicular)^{2}

Given: A triangle ABC right angled at B

To Prove: large AB^2+BC^2=AC^2

Construction: From B draw BD large perp AC

Proof: Since BD large perp AC

              large Delta ADBsimDelta ABC      [By the above theorem]

 large Rightarrow frac{AD}{AB}=frac{AB}{AC}             

large Rightarrow AB^2= AD;X;AC                           ............(1)

Now  large Delta BDCsimDelta ABC      [By the above theorem]

 large Rightarrow frac{BC}{AC}=frac{DC}{BC}                      

large Rightarrow BC^2= AC;X;DC                        ............(2)

Adding Eq (1) and (2)

large AB^2 +BC^2= AC;X;AD + AC;X;DC

large AB^2 + BC^2= AC;(AD + ;DC)

large AB^2 +BC^2= AC;X;AC

large AB^2 +BC^2= AC^2

Hence Proved

Example 1: The lengths of two sides of a right triangle are 5 cm and 12 cm. Find the length of the hypotenuse.

Solution:  Suppose BC = 5cm and AC = 12 cm.

             large therefore   By Pythagoras theorem,

                                large (AB)^{2}=(BC)^{2}+(AC)^{2}

                                large (AB)^{2}=(5)^{2}+(12)^{2}

                                                   large =25+144

                                                   large =169

                                 large (AB)^{2}=169

                          large ABtimes AB=13times 13

large Rightarrow                           large or;;AB=13 cm

large therefore   Length of the hypotenuse  = 13 cm.

Converse of Pythagoras Theorem:

THeorem: In a triangle if the square of one side is equal to the sum of squares of the other two sides, then the angle opposite to the first side is a right angle.

Given: A triangle ABC such that large AB^2+ BC^2=AC^2

To Prove : large angle B =90^0

Construction : Construct a triangle DEF such that DE= AB, EF= BC and large angle E =90^0

Proof: Since large Delta DEF is a right angled triangle with right angle at E

large therefore DF^2=DE^2+EF^2

Since DE = AB and EF = BC replacing them in the above equation

large therefore DF^2=AB^2+BC^2

But large AB^2+ BC^2=AC^2                    [Given]

large therefore DF^2=AC^2

   DF = AC

Now in large Delta ABC and large Delta DEF

  AB = DE             [By Construction]

 BC= EF               [By Construction ]

 DF = AC             [Proved Above]

large therefore Delta ABCcongDelta DEF                     [By SSS Congruence Criterion]

large Rightarrow angle B = angle E=90^0

Hence large Delta ABC is right angled at B

Illustration: ABC is an isosceles triangle with Ac = BC. If large AB^2=2AC^2. Prove that ABC is aright triangle.

In large Delta ABC, we are given that

AC = BC and large AB^2=2AC^2

Now large AC^2= BC^2

Adding large AC^2 To both sides we get

large AC^2+AC^2=BC^2+AC^2

large BC^2+AC^2=2AC^2

But large AB^2=2AC^2

large therefore BC^2+AC^2=AB^2

From Converse of Pythagoras Theorem we can say that

Triangle ABC is a right angled at C

 

Sample Questions
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Question : 1

In a right angle triangle ABC, right angled at B , AB = 6cm , BC = 8cm , then AC = _________________ .

Right Option : A
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Question : 2

If square of one side of triangle is equal to the sum of the square of other two sides then triangle is ______________________.

Right Option : C
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Explanation
Question : 3

If in large Delta ABC , AB = 6 cm, BC = 12cm and CA = large 6sqrt3 cm , then the measureof <A is _________________

Right Option : D
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Explanation
 
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