To determine the fourth vertex needed to create a rectangle given the vertices [tex]\((-7,1)\)[/tex], [tex]\((-7,-6)\)[/tex], and [tex]\((7,1)\)[/tex], follow these steps:
1. Identify the unpaired coordinates:
- The x-coordinates given are [tex]\(-7, -7,\)[/tex] and [tex]\(7\)[/tex]. Notice that [tex]\(-7\)[/tex] is used twice, meaning the x-coordinate for the fourth vertex must be [tex]\(7\)[/tex] to form a rectangle.
- The y-coordinates given are [tex]\(1, -6,\)[/tex] and [tex]\(1\)[/tex]. Notice that [tex]\(1\)[/tex] is used twice, meaning the y-coordinate for the fourth vertex must be [tex]\(-6\)[/tex] to form a rectangle.
2. Determine the missing coordinate:
- We observe that to complete the rectangle with one vertex having an x-coordinate of [tex]\(7\)[/tex] and the other with a y-coordinate of [tex]\(-6\)[/tex], the missing vertex will have coordinates [tex]\((7, -6)\)[/tex].
3. Verify alignment:
- Check if our calculated vertex [tex]\((7, -6)\)[/tex] aligns well with the other given vertices to form a proper rectangle.
- The vertices are:
- [tex]\((-7, 1)\)[/tex]
- [tex]\((-7, -6)\)[/tex]
- [tex]\((7, 1)\)[/tex]
- [tex]\((7, -6)\)[/tex]
- These vertices form parallel sides and right angles, characteristic of a rectangle.
Thus, the fourth vertex needed to create a rectangle with the given vertices is:
[tex]\[\boxed{(7, -6)}\][/tex]
So the correct option is:
[tex]\[
\boxed{(7, -6)}
\][/tex]
