Final answer:
To find the coordinates of the fourth vertex of a rectangle in a Cartesian coordinate system, analyze the known vertices and apply geometric properties of rectangles using the midpoint formula or displacement methods and verify the solution by checking that opposite sides are equal and parallel.
Step-by-step explanation:
Finding the fourth vertex of a rectangle given the other three involves understanding the properties of a rectangle and applying them to the Cartesian coordinate system. A rectangle has opposite sides that are parallel and equal in length. If the vertices of the rectangle in the Cartesian plane are A(x1, y1), B(x2, y2), C(x3, y3), we can denote the missing vertex as D(x4, y4).
To find the coordinates of D, you can use the midpoint formula or the concept of diagonal vectors being equal and bisecting each other. Assume that AB and CD are parallel, as are BC and AD. Without loss of generality, consider AB and CD as horizontal sides and AD and BC as the vertical sides. The coordinates of D can be determined by one of the following approaches:
- If AC is a diagonal, then the midpoint M1 of AC should also be the midpoint of BD. So use midpoint formula to find D.
- If you know one horizontal side, say AB, then you can calculate the change in the x direction (dx) and apply it to the known x coordinate of either A or B to get the x coordinate of D. Do the same with the change in the y direction (dy) for the vertical side.
Once the fourth vertex's coordinates are determined, always check the math to ensure the sides AB and CD, and AD and BC are parallel and of the same length, confirming a rectangle.