[[1]] in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.

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Accepted Answer

A right-angled triangle ABC which is right-angled at BTo prove:

A
    C
    2
   
   =A
    B
    2
   
   +B
    C
    2

Construction: Draw a perpendicular BD from B to ACProof:-In triangle ABC and ABD

∠ADB=∠ABC=
      90
      ∘
     
     (Given)

∠DAB=∠BAC=(common angle)

∴ΔABC∼ΔABD
      
       using AA similarity criteria

AA similarity criterion :- It states that if two triangles have two pairs of congruent angles, then the triangles are similar.Now, Applying properties of similar triangles, we have corresponding sides proportional to each other.

AD
      
       AB
     
     =
      
       AB
      
       AC

⇒A
      B
      2
     
     =AD×AC→eqn.1

Similarly,

∴ΔABC∼ΔBDC(using AA similarity criteria)
  Again, Applying properties of similar triangle, we have corresponding sides proportional to each other

BC
    
     CD
   
   =
    
     AC
    
     BC

⇒B
    C
    2
   
   =CD×AC→eqn.2
  Adding equation (1) and (2), we get

A
    B
    2
   
   +B
    C
    2
   
   =AD×AC+CD×AC
  Taking AC common, we get

A
    B
    2
   
   +B
    C
    2
   
   =AC(AD+CD)→eqn.3
  And from the figure we know that AD + CD = ACAfter changing the obtained value of AC above in equation (3), we get

A
    B
    2
   
   +B
    C
    2
   
   =AC×AC=A
    C
    2

in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.

[[1]] in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of two sides.

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