Prove that the points ( 2,3 ) , ( - 4 , - 6 ) and ( 1 , 3 2 ) do not form a triangle.

The given three points are A ( 2,3 ) , B ( - 4 , - 6 ) and C ( 1 , 3 2 ) . The distance d between two points x 1 , y 1 and ( x 2 , y 2 ) is given by the formula d = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 A B = ( 2 + 4 ) 2 + ( 3 + 6 ) 2 = 36 + 81 = 117 = 10 . 82 units B C = ( - 4 - 1 ) 2 + ( - 6 - 3 2 ) 2 = 25 + 225 4 = 81.25 = 9 . 01 units A C = ( 2 - 1 ) 2 + ( 3 - 3 2 ) 2 = 1 + 9 4 = 3 . 25 = 1 . 8 units Since, 9 . 01 + 1 . 8 < 10 . 82 Therefore, B C + A C ≯ A B This is in violation of the basic property of any triangle to exist. Hence, the given three points do not form a triangle.

Prove that the points ( 2,3 ) , ( - 4 , - 6 ) and ( 1 , 3 2 ) do not form a triangle.

The given three points are A ( 2,3 ) , B ( - 4 , - 6 ) and C ( 1 , 3 2 ) . The distance d between two points x 1 , y 1 and ( x 2 , y 2 ) is given by the formula d = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 A B = ( 2 + 4 ) 2 + ( 3 + 6 ) 2 = 36 + 81 = 117 = 10 . 82 units B C = ( - 4 - 1 ) 2 + ( - 6 - 3 2 ) 2 = 25 + 225 4 = 81.25 = 9 . 01 units A C = ( 2 - 1 ) 2 + ( 3 - 3 2 ) 2 = 1 + 9 4 = 3 . 25 = 1 . 8 units Since, 9 . 01 + 1 . 8 < 10 . 82 Therefore, B C + A C ≯ A B This is in violation of the basic property of any triangle to exist. Hence, the given three points do not form a triangle.