If (-1, -1) and (3, -1) are two opposite corners of a square, the other two corners a

Options

A. (2, 0), (- 2, 2)
B. (2, -2), (0, 2)
C. (3,0), (4, - 2)
D. None of these

Answer

To find the other two corners of the square given the opposite corners (-1, -1) and (3, -1), we will follow these steps:

### Step 1: Identify the given points
The two opposite corners of the square are:
- Point A: (-1, -1)
- Point B: (3, -1)

### Step 2: Find the midpoint of the diagonal
The midpoint of the diagonal can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ \text{Midpoint} = \left( \frac{-1 + 3}{2}, \frac{-1 + (-1)}{2} \right) = \left( \frac{2}{2}, \frac{-2}{2} \right) = (1, -1) \]

### Step 3: Determine the slope of the line segment AB
The slope of the line segment connecting points A and B can be calculated as:
\[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-1)}{3 - (-1)} = \frac{0}{4} = 0 \]
This indicates that the line segment AB is horizontal.

### Step 4: Find the coordinates of the other two corners
Since the square is oriented such that its sides are perpendicular to the line segment AB, the other two corners will be vertically aligned with the midpoint. The distance from the midpoint to each corner is equal to half the length of the diagonal.

The length of the diagonal AB can be calculated as:
\[ \text{Length of AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - (-1))^2 + (-1 - (-1))^2} = \sqrt{(4)^2 + (0)^2} = 4 \]
Thus, half the length of the diagonal is 2.

### Step 5: Calculate the coordinates of the other corners
The other two corners will be located 2 units above and below the midpoint (1, -1):
- Corner C: (1, -1 + 2) = (1, 1)
- Corner D: (1, -1 - 2) = (1, -3)

### Final Answer
The other two corners of the square are:
- Corner C: (1, 1)
- Corner D: (1, -3)

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