If (1, 1) and (1, 8) are the opposite vertices of a square, then find the co-oridnates of remaining two vertices.

Answer

To find the coordinates of the remaining two vertices of the square given the opposite vertices (1, 1) and (1, 8), we can follow these steps:

### Step 1: Identify the Given Points
Let the given points be:
- A (1, 1)
- C (1, 8)

### Step 2: Calculate the Length of the Diagonal
The distance between points A and C can be calculated using the distance formula:
\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and C:
\[ AC = \sqrt{(1 - 1)^2 + (8 - 1)^2} = \sqrt{0 + 49} = 7 \]
The length of the diagonal AC is 7 units.

### Step 3: Find the Midpoint of AC
The midpoint M of the diagonal AC can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + 1}{2}, \frac{1 + 8}{2} \right) = (1, 4.5) \]

### Step 4: Determine the Slope of AC
The slope of line AC is given by:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 1}{1 - 1} = \text{undefined} \]
Since the line is vertical, the slope of the line perpendicular to AC (which will be the sides of the square) will be horizontal.

### Step 5: Calculate the Length of Each Side
Since the diagonal of the square is \(7\), the length of each side \(s\) can be calculated using the relationship:
\[ s = \frac{AC}{\sqrt{2}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2} \]

### Step 6: Find the Remaining Vertices
To find the coordinates of the remaining vertices B and D, we can move horizontally from the midpoint M (1, 4.5) by \( \frac{7\sqrt{2}}{2} \) units to the left and right.

1. **Vertex B** (moving left):
\[ B = \left( 1 - \frac{7\sqrt{2}}{4}, 4.5 \right) \]

2. **Vertex D** (moving right):
\[ D = \left( 1 + \frac{7\sqrt{2}}{4}, 4.5 \right) \]

### Step 7: Final Coordinates
Thus, the coordinates of the remaining two vertices are:
- B: \( \left( 1 - \frac{7\sqrt{2}}{4}, 4.5 \right) \)
- D: \( \left( 1 + \frac{7\sqrt{2}}{4}, 4.5 \right) \)