Krishna
0

Step 1: Recall the distance between two points formula

             NOTE: Measure the distance between the points. to know the lengths

            of the three sides of the triangle.

              FORMULA: d\ =\ \sqrt{\left(x_{2\ }-\ x_1\right)^2\ +\ \left(y_{2\ }\ -\ y_1\right)^2}\   

            

Step 2: Find the tree lengths of the equilateral triangle. Using the distance between the two points formula.

             NOTE: In equilateral triangle  all sides are equal

             EXAMPLE:  AC = \sqrt{(x - 0)^2 + (y - 0)^2} = 12

                                 AC = \sqrt{x^2 + y^2} = 12 (given)

                                 AC =   x^2 + y^2 = 12^2.................(1)


                                 AB = \sqrt{b^2} = 12

                                                 b = 12..........................(2)


                                BC = \sqrt{(x - b)^2 + (y - 0)^2 } = 12

                                         \sqrt{(x - b)^2 + (y)^2 } = 12 ................(3)


Step 3: Find the x value by simplifying the equations

            NOTE: Solve the equations in step 2

            EXAMPLE: Substitute the b value in the equation (3)

                               \sqrt{(x - 12)^2 + (y)^2}  = 12

                               (x)^2 - 2*12*x + (12)^2 + (y)^2 = (12)^2    

                               x^2 + y^2 - 24x + (12)^2 = (12)^2

                            We can x^2 + y^2 = 12^2 since equation (1)

                               12^2 - 24x + 12^2 = 12^2

                               - 24x = 12^2

                               x = - \frac{144}{24}  

                                          x = - 6

Step 4: Substitute x value in the equation (1) to find the y value.

                   EXAMPLE: x^2 + y^2 = 12^2

                                       (-6)^2 + y^2 = 12^2

                                       36 + y^2 = 144

                                       y^2 = 108    

                                       y = 6 \sqrt{3}          

           Coordinates of C = (-6, 6 \sqrt{3} )