Fill in the blank.If the vertices of a triangle have integral coordinates, the triangle cannot be ____.

Question

Accepted Answer

If the vertices of a triangle have integral coordinates, the triangle cannot be equilateral.It is given that Vertices of triangle have integral coordinates.Let, The coordinates of triangles are

A
    
     0,0
   B
    
     x,y
   C
    
     a,b
   
  .We know that,The slope of a line is given as

m=

y
      2
     
     −
      y
      1

x
      2
     
     −
      x
      1

.Then the slope of AB and AC will be

Slope of AB, 
      m
      1
     
     =
      y
      x
     
     ,

And

Slope of AC, 
      m
      2
     
     =
      b
      a
     
     .

From above,Both being rational.Since, If m1 and m2 are the slopes of two lines, then the angle between them is written as,

tanθ=

m
      1
     
     −
      m
      2

1+
      m
      1

m
      2

. Then, the angle between AB and AC will be

tanθ=

y
      x
     
     −
      b
      a

1+
      y
      x
     
     ×
      b
      a

which is rational.At

tan
    
     60
    °
   
   =
    3

, there cannot be any point with rational coefficient. So, angle between AB and AC cannot be 60°.Therefore,It is impossible to obtain an equilateral triangle since there are no rational points on the line that are at a 60° angle.Hence, if the vertices of a triangle have integral coordinates, the triangle cannot be equilateral.

Fill in the blank.

If the vertices of a triangle have integral coordinates, the triangle cannot be ____.

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