(1,2) and (3,8) are a pair of a opposite vertices of square. Find the diagonals of the square passing through (1,2) :

Options

A. a. x-2y =1
B. b. 2x+7y =0
C. c. 3x+2y +7 =0
D. d. 3x-y=1

Answer

To find the diagonals of the square passing through the point (1, 2) given the opposite vertex (3, 8), we can follow these steps:

### Step 1: Identify the coordinates of the opposite vertices
The opposite vertices of the square are given as (1, 2) and (3, 8). Let's denote these points as A(1, 2) and C(3, 8).

### Step 2: Find the midpoint of the diagonal AC
The midpoint M of the diagonal AC 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 A and C:
\[ M = \left( \frac{1 + 3}{2}, \frac{2 + 8}{2} \right) = \left( \frac{4}{2}, \frac{10}{2} \right) = (2, 5) \]

### Step 3: Calculate the slope of the diagonal AC
The slope \( m \) of the line passing through points A and C can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and C:
\[ m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3 \]

### Step 4: Find the slope of the perpendicular diagonal
The slope of the diagonal perpendicular to AC (let's call it BD) is the negative reciprocal of the slope of AC:
\[ m_{BD} = -\frac{1}{m} = -\frac{1}{3} \]

### Step 5: Use point-slope form to find the equation of diagonal BD
Using the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
We can use point A(1, 2) and the slope \( m_{BD} = -\frac{1}{3} \):
\[ y - 2 = -\frac{1}{3}(x - 1) \]
Expanding this:
\[ y - 2 = -\frac{1}{3}x + \frac{1}{3} \]
Adding 2 to both sides:
\[ y = -\frac{1}{3}x + \frac{1}{3} + 2 \]
\[ y = -\frac{1}{3}x + \frac{7}{3} \]

### Step 6: Rearranging to standard form
To rearrange the equation into standard form:
\[ \frac{1}{3}x + y - \frac{7}{3} = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ x + 3y - 7 = 0 \]

### Final Answer
The equation of the diagonal passing through (1, 2) is:
\[ x + 3y - 7 = 0 \]