The coordinates of the three vertices of a square ABCD are A (-3, 5), B (1, 7), and C (3, 3).

What are the coordinates of vertex D?
Answer
A.(-4, 2)
B.(-2, 1)
C.(-1, 1)
D.(-4, -2)

To find the coordinates of vertex D, we need to examine the properties of a square.

A square is a special type of quadrilateral where all four sides are equal in length and all four angles are right angles (90 degrees).

In this given problem, we know the coordinates of three vertices, A, B, and C. To find the coordinates of vertex D, we need to determine the lengths of the sides of the square using the given coordinates.

First, let's find the length of side AB:

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

Substituting the coordinates of A (-3, 5) and B (1, 7), we get:

Distance AB = sqrt((1 - (-3))^2 + (7 - 5)^2)
= sqrt((1 + 3)^2 + (7 - 5)^2)
= sqrt((4)^2 + (2)^2)
= sqrt(16 + 4)
= sqrt(20)

Next, let's find the length of side BC:

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

Substituting the coordinates of B (1, 7) and C (3, 3), we get:

Distance BC = sqrt((3 - 1)^2 + (3 - 7)^2)
= sqrt((2)^2 + (-4)^2)
= sqrt(4 + 16)
= sqrt(20)

Since all sides of a square are equal in length, we have:

AB = BC = sqrt(20)

Finally, let's find the coordinates of vertex D by using the distance and direction from vertex C:

To go from (3, 3) to (x, y), we need to move a distance of sqrt(20) units in the opposite direction of the line BC.

To move in the opposite direction, we subtract the distance from C to D.
Thus,

x-coordinate of D = 3 - sqrt(20)
y-coordinate of D = 3 - sqrt(20)

Therefore, the coordinates of vertex D are (-3 - sqrt(20), 3 - sqrt(20)).

Now you can check which of the given options matches the coordinates we derived.