If (1, 2) and (3, 8) are a pair of opposite vertices of a square, find the equation of the sides and diagonals of the square.

Answer

To find the equations of the sides and diagonals of a square given the opposite vertices (1, 2) and (3, 8), we can follow these steps:

### Step 1: Identify the given vertices
Let the vertices of the square be:
- A(1, 2)
- C(3, 8)

### Step 2: Find the midpoint of the diagonal AC
The midpoint O of the diagonal AC can be calculated using the midpoint formula:
\[ O\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of A and C:
\[ O\left(\frac{1 + 3}{2}, \frac{2 + 8}{2}\right) = O(2, 5) \]

### Step 3: Calculate the length of the diagonal AC
Using the distance formula, the length of diagonal AC is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and C:
\[ d = \sqrt{(3 - 1)^2 + (8 - 2)^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]

### Step 4: Find the length of the side of the square
The diagonal of a square is related to its side length \( s \) by the formula:
\[ d = s\sqrt{2} \]
Thus, we can find the side length:
\[ s = \frac{d}{\sqrt{2}} = \frac{2\sqrt{10}}{\sqrt{2}} = \sqrt{20} = 2\sqrt{5} \]

### Step 5: Find the coordinates of the other two vertices B and D
The slope of the line AC can be calculated as:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3 \]
The slope of the sides of the square will be perpendicular to this slope. Therefore, the slope of lines AB and CD will be:
\[ \text{slope}_{AB} = -\frac{1}{3} \]

Using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), we can find the equations of lines AB and CD:
1. For point A(1, 2):
\[ y - 2 = -\frac{1}{3}(x - 1) \implies 3y - 6 = -x + 1 \implies x + 3y = 7 \quad \text{(Equation 1)} \]
2. For point C(3, 8):
\[ y - 8 = -\frac{1}{3}(x - 3) \implies 3y - 24 = -x + 3 \implies x + 3y = 27 \quad \text{(Equation 2)} \]

### Step 6: Find the coordinates of B and D
To find the coordinates of B and D, we can use the fact that they are at a distance of \( 2\sqrt{5} \) from the midpoint O(2, 5) in the direction perpendicular to AC.

Using the direction vector of AC, which is (2, 6), we can find the unit vector and then scale it to find B and D.

### Step 7: Write the equations of the diagonals
The diagonals of the square are AC and BD. The equations of the diagonals can be derived similarly:
1. For diagonal AC (from A to C):
\[ y - 2 = 3(x - 1) \implies y = 3x - 1 \quad \text{(Diagonal 1)} \]
2. For diagonal BD (using the coordinates of B and D):
The slope will be the negative reciprocal of the slope of AC, and you can find the equation similarly.

### Final Result
The equations of the sides and diagonals of the square are:
- Side AB: \( x + 3y = 7 \)
- Side CD: \( x + 3y = 27 \)
- Diagonal AC: \( y = 3x - 1 \)
- Diagonal BD: (derived from coordinates of B and D)