What are the coordinates of the vertices of quadrilateral XYZD if the coordinates of the vertices of quadrilateral PQRS are P (-10, 10), Q (-2, 10), R (-2, 2), S (-10, 2), and Y (0, 0) is the midpoint of segment PR?

To find the coordinates of the vertices of quadrilateral XYZD, we need to determine the coordinates of point Y and use that information to find the coordinates of the other vertices.

We are given that Y (0, 0) is the midpoint of segment PR. This means that the average of the x-coordinates of P and R is zero, and the average of the y-coordinates of P and R is also zero.

Let's find the coordinates of P and R first.

The x-coordinate of P is -10, and the x-coordinate of R is -2. To find the average x-coordinate, we add the x-coordinates of P and R and divide by 2:
(-10 + -2) / 2 = -12 / 2 = -6

Similarly, the y-coordinate of P is 10, and the y-coordinate of R is 2. To find the average y-coordinate, we add the y-coordinates of P and R and divide by 2:
(10 + 2) / 2 = 12 / 2 = 6

So, the midpoint Y (0, 0) is the average of the x-coordinates and the y-coordinates of P and R.

Now that we know the average coordinates, we can find the coordinates of X, Z, and D.

Since the x-coordinate of Y is 0, the x-coordinate of X should be twice the x-coordinate of Y. Similarly, the y-coordinate of X should be twice the y-coordinate of Y.

Therefore, the coordinates of X are (2 * 0, 2 * 0) = (0, 0).

Similarly, the x-coordinate of Z should be twice the x-coordinate of Y, and the y-coordinate of Z should be the same as the y-coordinate of R.

Therefore, the coordinates of Z are (2 * 0, 2) = (0, 2).

Lastly, the x-coordinate of D should be the same as the x-coordinate of P, and the y-coordinate of D should be twice the y-coordinate of Y.

Therefore, the coordinates of D are (-10, 2 * 0) = (-10, 0).

Hence, the coordinates of the vertices of quadrilateral XYZD are:
X (0, 0)
Y (0, 0)
Z (0, 2)
D (-10, 0)