Final answer:
To find the coordinates of point D in rectangle ABCD, we first determine the coordinates of B and C using the midpoints provided, then use the properties of a rectangle to deduce that D must have the same x-coordinate as A, and the same y-coordinate as C. Consequently, the coordinates for point D are (4, -8).
Step-by-step explanation:
The objective is to find the coordinates of point D of rectangle ABCD given certain information about the midpoints of sides AB and BC. Since point M (1, 2) is the midpoint of AB, and we know point A (4,2), we can find point B by understanding that the x-coordinate of the midpoint is the average of the x-coordinates of A and B. So, if M is the midpoint, 1 = (4 + xB)/2; solving this gives us xB = -2. The y-coordinate of B must be the same as that of A and M because AB is horizontal in a rectangle, so yB = 2.
Using the same logic, since point N (-2, -3) is the midpoint of BC and point B (-2, 2), we can find point C by considering that C must have the same x-coordinate as B in a rectangle because BC is vertical, so xC = -2. For the y-coordinate, if N is the midpoint, -3 = (2 + yC)/2; solving this gives us yC = -8.
To find point D, since AB and CD are parallel and equal in a rectangle, the x-coordinate of D must be the same as that of A which is 4. The y-coordinate must be the same as C, which is -8. Therefore, the coordinates of point D are (4, -8).