ABCD is a parallelogram. Co-ordinates of A, B and C are (5,0), (-2,3) and (-1,4) respectively. What will be the equation of line AD?

Coordinates of A (5,0) , B (-2,3) and C (-1,4). Let coordinates of D = (x,y)

ABCD is a parallelogram and AB is parallel to CD, thus slopes of AB and CD are equal.

=> $$\frac{3 - 0}{-2 - 5} = \frac{4 - y}{-1 - x}$$

=> $$\frac{-3}{7} = \frac{4 - y}{-1 - x}$$

=> $$3 + 3x = 28 - 7y$$

=> $$3x + 7y = 25$$ -----------------(i)

Also, BC is parallel to AD

=> $$\frac{3 - 4}{-2 + 1} = \frac{y - 0}{x - 5}$$

=> $$x - 5 = y$$ ------------------(ii)

Substituting value of 'y' in equation (i), we get : $$3x + 7(x - 5) = 25$$

=> $$3x + 7x - 35 = 25$$

=> $$x = \frac{60}{10} = 6$$

Putting it in equation (ii), => $$y = 6 - 5 = 1$$

Now, equation of line AD with coordinates (5,0) and (6,1)

=> $$(y - 0) = \frac{1 - 0}{6 - 5} (x - 5)$$

=> $$y = x - 5$$