The vertices of a diagonal of a square are `(-2,4)` and `(-2,-2)` Find the other vertices

Options

A. (1,-1),(5,1)
B. (1,1),(5,-1)
C. (1,1),(-5,1)
D. none of these

Answer

To find the other vertices of the square given the vertices of a diagonal, we can follow these steps:

### Step 1: Identify the given points
The vertices of the diagonal of the square are given as:
- Point A: (-2, 4)
- Point C: (-2, -2)

### Step 2: Calculate the midpoint of the diagonal
The midpoint \( M \) of the diagonal can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and C:
\[ M = \left( \frac{-2 + (-2)}{2}, \frac{4 + (-2)}{2} \right) = \left( -2, 1 \right) \]

### Step 3: Determine the distance of the diagonal
The distance \( AC \) can be calculated using the distance formula:
\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and C:
\[ AC = \sqrt{((-2) - (-2))^2 + ((-2) - 4)^2} = \sqrt{0 + (-6)^2} = 6 \]

### Step 4: Calculate the length of the sides of the square
Since the diagonal of a square relates to its side length \( s \) by the formula:
\[ AC = s\sqrt{2} \]
We can solve for \( s \):
\[ s\sqrt{2} = 6 \implies s = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Step 5: Determine the other vertices
Let the other two vertices of the square be \( B(x_1, y_1) \) and \( D(x_2, y_2) \). Since the square is symmetric about the midpoint, we can find the coordinates of points B and D using the properties of the square.

Using the fact that the slope of the diagonal AC is undefined (vertical line), the other two vertices B and D will be horizontally aligned with the midpoint M. The coordinates of B and D can be calculated as:
- \( B = (-2 + \frac{s}{2}, 1) = (-2 + \frac{3\sqrt{2}}{2}, 1) \)
- \( D = (-2 - \frac{s}{2}, 1) = (-2 - \frac{3\sqrt{2}}{2}, 1) \)

Calculating these:
1. For point B:
\[ B = \left(-2 + \frac{3\sqrt{2}}{2}, 1\right) \]
2. For point D:
\[ D = \left(-2 - \frac{3\sqrt{2}}{2}, 1\right) \]

### Step 6: Final coordinates of all vertices
Thus, the coordinates of the vertices of the square are:
- A: (-2, 4)
- B: \((-2 + \frac{3\sqrt{2}}{2}, 1)\)
- C: (-2, -2)
- D: \((-2 - \frac{3\sqrt{2}}{2}, 1)\)