Which set of coordinates, when paired with \((-3, -2)\) and \((-5, -2)\), result in a square?

A. \((-5, -2)\) and \((-3, -2)\)
B. \((-3, 0)\) and \((0, -2)\)
C. \((-3, 0)\) and \((-5, 0)\)
D. [tex]\((0, -2)\)[/tex] and [tex]\((0, -3)\)[/tex]



Answer :

To determine which set of coordinates, when paired with [tex]$(-3, -2)$[/tex] and [tex]$(-5, -2)$[/tex], forms a square, we need to analyze each set of additional coordinates. The essential properties of a square are equal side lengths and equal diagonals.

1. First Pair: [tex]$(-3,0)$[/tex] and [tex]$(0,-2)$[/tex]:
- Given points: [tex]$(-3, -2)$[/tex], [tex]$(-5, -2)$[/tex]
- Additional points: [tex]$(-3, 0)$[/tex], [tex]$(0, -2)$[/tex]

2. Second Pair: [tex]$(-3,0)$[/tex] and [tex]$(-5,0)$[/tex]:
- Given points: [tex]$(-3, -2)$[/tex], [tex]$(-5, -2)$[/tex]
- Additional points: [tex]$(-3, 0)$[/tex], [tex]$(-5, 0)$[/tex]

3. Third Pair: [tex]$(0,-2)$[/tex] and [tex]$(0,-3)$[/tex]:
- Given points: [tex]$(-3, -2)$[/tex], [tex]$(-5, -2)$[/tex]
- Additional points: [tex]$(0, -2)$[/tex], [tex]$(0, -3)$[/tex]

To check if these points form a square, we measure the distances between all pairs of points in each set. But let's evaluate based on the geometric relationship and properties mathematically.

After evaluating all sets of coordinates:

- Set 1: [tex]$(-3, 0)$[/tex] and [tex]$(0, -2)$[/tex] does not form a square.
- Set 2: [tex]$(-3, 0)$[/tex] and [tex]$(-5, 0)$[/tex] does not form a square.
- Set 3: [tex]$(0, -2)$[/tex] and [tex]$(0, -3)$[/tex] does not form a square.

None of these sets, when paired with [tex]$(-3, -2)$[/tex] and [tex]$(-5, -2)$[/tex], result in a square. Therefore, there isn't a set of the given coordinates that can produce a square with the initial two points.

The answer is:
[tex]\[ \boxed{\text{None}} \][/tex]