In a rhombus ABCD the diagonals AC and BD intersect at the point (3,4) . If the point A is (1,2) the diagonal BD has the equation

Options

A. x - y - 1 = 0
B. x + y - 1 = 0
C. x - y + 1 = 0
D. x + y - 7 =0

Answer

To find the equation of the diagonal BD in the rhombus ABCD, we will follow these steps:

### Step 1: Identify the given points
We are given:
- The intersection point of the diagonals (which is also the midpoint of both diagonals) is \( (3, 4) \).
- The coordinates of point A are \( (1, 2) \).

### Step 2: Calculate the slope of diagonal AC
The slope \( m_1 \) of line AC can be calculated using the formula for the slope between two points:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can take point A as \( (x_1, y_1) = (1, 2) \) and the intersection point as \( (x_2, y_2) = (3, 4) \):
\[ m_1 = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1 \]

### Step 3: Determine the slope of diagonal BD
Since the diagonals of a rhombus are perpendicular to each other, the slope \( m_2 \) of diagonal BD can be found using the relationship:
\[ m_1 \cdot m_2 = -1 \]
Substituting \( m_1 = 1 \):
\[ 1 \cdot m_2 = -1 \implies m_2 = -1 \]

### Step 4: Use the point-slope form to find the equation of line BD
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -1 \) and the point \( (3, 4) \):
\[ y - 4 = -1(x - 3) \]
Expanding this:
\[ y - 4 = -x + 3 \]
Rearranging gives:
\[ x + y - 7 = 0 \]

### Step 5: Final equation of diagonal BD
Thus, the equation of diagonal BD is:
\[ x + y - 7 = 0 \]