Prove that the points A ( 2 , 4 ) , B ( 2 , 6 ) and C ( 2 + 3 , 5 ) are the vertices of an equilateral triangle.

The given points are A ( 2 , 4 ) , B ( 2 , 6 ) and C 2 + 3 , 5 . Now, A B = ( 2 - 2 ) 2 + ( 4 - 6 ) 2 = ( 0 ) 2 + ( - 2 ) 2 = ( 0 + 4 ) = 2 units B C = 2 - 2 - 3 2 + ( 6 - 5 ) 2 = - 3 2 + 1 2 = 3 + 1 = 2 units A C = 2 - 2 - 3 2 + ( 4 - 5 ) 2 = - 3 2 + 1 2 = 3 + 1 = 2 units Therefore, the points A ( 2 , 4 ) , B ( 2 , 6 ) and C 2 + 3 , 5 are the vertices of an equilateral triangle.

Show that the points ( - 3 , - 3 ) , ( 3 , 3 ) and - 3 3 , 3 3 are the vertices of an equilateral triangle.

Suppose the given points be A ( - 3 , - 3 ) , B ( 3 , 3 ) and C - 3 3 , 3 3 Now, A B = ( - 3 - 3 ) 2 + ( - 3 - 3 ) 2 = ( - 6 ) 2 + ( - 6 ) 2 = 36 + 36 = 72 = 6 2 units B C = 3 + 3 3 2 + 3 - 3 3 2 = 9 + 27 + 18 3 + 9 + 27 - 18 3 = 72 = 6 2 units A C = - 3 + 3 3 2 + - 3 - 3 3 2 = 3 - 3 3 2 + 3 + 3 3 2 = 9 + 27 - 18 3 + 9 + 27 + 18 3 = 72 = 6 2 units Therefore, the given points are the vertices of an equilateral triangle.

Show that the points O ( 0 , 0 ) , A 3 , 3 and B 3 , - 3 are the vertices of an equilateral triangle. Find the area of this triangle.

The given points are O ( 0 , 0 ) , A 3 , 3 and B 3 , - 3 . O A = ( 3 - 0 ) 2 + 3 ) - 0 2 = ( 3 ) 2 + 3 2 ) = 9 + 3 = 12 = 2 3 units A B = ( 3 - 3 ) 2 + - 3 - 3 2 = 0 + 2 3 2 = 4 3 = 12 = 2 3 units O B = 3 - 0 2 + - 3 - 0 2 = 3 2 + 3 2 = 9 + 3 = 12 = 2 3 units Therefore, O A = A B = O B = 2 3 units Thus, the points O 0 , 0 , A 3 , 3 and B 3 , - 3 are the vertices of an equilateral triangle. Also, the area of the triangle O A B = 3 4 × side 2 = 3 4 × 2 3 2 square unit = 3 4 × 12 square unit = 3 3 square units