If the three vertices of a rectangle taken in order are the points (2, -2), (8, 4) and (5, 7). the co-ordinates of the fourth vertex is

(-1, 1)

Step-by-step explanation:

Let D represent the fourth vertex with coordinate (x, y)  

Since ABCD is a rectangle, then BC || AD and AB || DC

Since BC is parallel to AD, then they have the same slope (m) = -1

Equation of line AD: y - (-2)) = -1(x - 2)   ⇒   y + 2 = -x + 2   ⇒   y = -x

Since AB is parallel to DC, then they have the same slope (m) = 1

Equation of line DC: y - 7 = 1(x - 5)   ⇒   y - 7 = x - 5   ⇒   y = x + 2

D is where the two lines intersect so the next step is to solve the system of equations:    y = -x     and     y = x + 2

Using substitution method: -x = x + 2   ⇒   -2x = 2   ⇒   x = -1

Now solve for y:  y = -x   ⇒   y = -(-1)   ⇒   y = 1

x = -1   and y = 1