ABCD is a rhombus. The co-ordinates of A and Care (3, 6) and (-1, 2) respectively. Find the equation of BD.

Answer

To find the equation of the diagonal BD of the rhombus ABCD given the coordinates of points A and C, we will follow these steps:

### Step 1: Identify the Coordinates of Points A and C
We are given:
- A(3, 6)
- C(-1, 2)

### Step 2: Find the Midpoint O of AC
The midpoint O of the line segment 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:
\[ O\left(\frac{3 + (-1)}{2}, \frac{6 + 2}{2}\right) = O\left(\frac{2}{2}, \frac{8}{2}\right) = O(1, 4) \]

### Step 3: Calculate the Slope of AC
The slope \( m_1 \) of line AC can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A and C:
\[ m_1 = \frac{2 - 6}{-1 - 3} = \frac{-4}{-4} = 1 \]

### Step 4: Find the Slope of BD
Since the diagonals of a rhombus are perpendicular, the slope \( m_2 \) of line BD is the negative reciprocal of the slope of AC:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{1} = -1 \]

### Step 5: Use the Point-Slope Form to Find the Equation of BD
We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using point O(1, 4) and slope \( m_2 = -1 \):
\[ y - 4 = -1(x - 1) \]
Simplifying this:
\[ y - 4 = -x + 1 \]
\[ y = -x + 5 \]

### Step 6: Rearranging to Standard Form
To express the equation in standard form:
\[ x + y = 5 \]

### Final Answer
The equation of diagonal BD is:
\[ x + y = 5 \]

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