Points W(-5,3), X(-1,6), and Y(5, -2) are three vertices of a rectangle. Determine the coordinates of the fourth vertex.

To determine the coordinates of the fourth vertex of the rectangle, we need to understand the properties of a rectangle.

A rectangle is a parallelogram with four right angles. This means that opposite sides of a rectangle are equal in length and parallel to each other. Also, the diagonals of a rectangle are equal in length and bisect each other.

In this case, we can start by finding the lengths of the sides of the given rectangle using the coordinates of the three vertices.

The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distances between the given points:
- Distance between W(-5, 3) and X(-1, 6)
Distance = sqrt((-1 - (-5))^2 + (6 - 3)^2)
= sqrt((4)^2 + (3)^2)
= sqrt(16 + 9)
= sqrt(25)
= 5

- Distance between W(-5, 3) and Y(5, -2)
Distance = sqrt((5 - (-5))^2 + (-2 - 3)^2)
= sqrt((10)^2 + (-5)^2)
= sqrt(100 + 25)
= sqrt(125)
= 5√5

- Distance between X(-1, 6) and Y(5, -2)
Distance = sqrt((5 - (-1))^2 + (-2 - 6)^2)
= sqrt((6)^2 + (-8)^2)
= sqrt(36 + 64)
= sqrt(100)
= 10

Since opposite sides of a rectangle are equal in length, we know that the fourth side (ZW) must also have a length of 5. Therefore, we can conclude that the fourth vertex (Z) must be located 5 units away from point X in the opposite direction.

To find the coordinates of Z, we start at point X(-1, 6) and move 5 units in the opposite direction of the vector from X to W.

Let's calculate the coordinates of Z:
- x-coordinate of Z: x-coordinate of X minus 5
x-coordinate of Z = -1 - 5 = -6

- y-coordinate of Z: y-coordinate of X minus 5
y-coordinate of Z = 6 - 5 = 1

Therefore, the coordinates of the fourth vertex Z are (-6, 1).

Draw it !!!!! call missing point P

slope XY = (-2-6)/(5+1) = -8/6 =-4/3
so
for XY
y = -(4/3) x + something
BUT
WP has same slope
y = -(4/3) x + b
3 = -(4/3) (-5) + b
b = 3 - 20/3 = -11/3
3 y = -4x - 11 is WP
Find YP the same way it is parallel to XW
then
find intersection of YP and WP