To find the coordinates of the fourth vertex of a rectangle when three of its vertices are given, we can follow these steps:
1. Identify Given Points:
The vertices given are:
- [tex]\( A = (-2, 3) \)[/tex]
- [tex]\( B = (4, 3) \)[/tex]
- [tex]\( C = (4, -3) \)[/tex]
2. Understand the Properties of a Rectangle:
In a rectangle, opposite sides are parallel and equal in length, and all interior angles are right angles (90 degrees).
3. Check Alignment of Given Points:
- Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have the same [tex]\( y \)[/tex]-coordinate, which means the line segment [tex]\( AB \)[/tex] is horizontal.
- Points [tex]\( B \)[/tex] and [tex]\( C \)[/tex] have the same [tex]\( x \)[/tex]-coordinate, meaning the line segment [tex]\( BC \)[/tex] is vertical.
4. Determine Missing Coordinates:
- We need to find point [tex]\( D \)[/tex] such that [tex]\( D \)[/tex] forms a rectangle with the given points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex].
- For [tex]\( D \)[/tex] to align with the properties of a rectangle, it must share the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] and the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex].
5. Combine Identified Coordinates:
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( D \)[/tex] should match the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex]: [tex]\( -2 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of point [tex]\( D \)[/tex] should match the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex]: [tex]\( -3 \)[/tex].
Therefore, the coordinates of the fourth vertex [tex]\( D \)[/tex] are:
[tex]\[
(-2, -3)
\][/tex]
