Is it true that the points (0,0),(5,5) and (-5,5) are the vertices of a right angled isosceles triangle?

Question

Accepted Answer

Given data,Three points (0,0),(5,5) and (-5,5) are the vertices of a right angled isosceles triangle.We know, from Pythagorean theorem

Q
    R
    2
   
   =P
    Q
    2
   
   +P
    R
    2

. Using the distance formula D=x2-x12+y2-y12. An isosceles triangle has two sides equal.Let the vertices of the triangle be,

A=
    
     0,0
   ,B=
    
     5,5
   ,C=
    
     −5,5
   
  Now, substituting x2=5, x1=0, y2=5, and y1=0, in the formula,

AB=

(5−0)
      2
     
     +
      
       (5−0)
      2

=
    
     25+25
   
   =5
    2

Substituting x2=-5, x1=0, y2=5, and y1=0, in the formula,

AC=

(−5−0)
      2
     
     +
      
       (5−0)
      2

=
    
     25+25
   
   =5
    2

Substituting x2=5, x1=-5, y2=5, and y1=-5, in the formula,

BC=

(5+5)
      2
     
     +
      
       (5−5)
      2

=
    
     100
   
   =10
  Therefore,

AB=AC
   Hence, the triangle is isosceles. Now,

A
    B
    2
   
   +A
    C
    2
   
   =
    
     (5
      2
     
     )
    2
   
   +
    
     (5
      2
     
     )
    2
   
   =100
  And,

B
    C
    2
   
   =
    
     (10)
    2
   
   =100

∴A
    B
    2
   
   +A
    C
    2
   
   =B
    C
    2

As a result, the Pythagorean theorem was also satisfied.

∴A=
    
     90
    °

and

ΔABC
  is a right-angled isosceles triangleTherefore, our statement is true.Hence, option 1 is correct.

Is it true that the points (0,0),(5,5) and (-5,5) are the vertices of a right angled isosceles triangle?

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