B (-5, 6) and D (1, 4) are the vertices of rhombus ABCD. Find the equations of diagonals BD and AC. 

Answer

To find the equations of the diagonals BD and AC of rhombus ABCD, where the vertices B and D are given as B(-5, 6) and D(1, 4), we can follow these steps:

### Step 1: Find the midpoint O of diagonal BD
The midpoint O of a line segment connecting points (x1, y1) and (x2, y2) is given by the formula:
\[ O = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
Substituting the coordinates of B and D:
\[ O = \left( \frac{-5 + 1}{2}, \frac{6 + 4}{2} \right) = \left( \frac{-4}{2}, \frac{10}{2} \right) = (-2, 5) \]

### Step 2: Find the slope of diagonal BD
The slope (m) of a line through points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]
Using points B(-5, 6) and D(1, 4):
\[ m_{BD} = \frac{4 - 6}{1 - (-5)} = \frac{-2}{1 + 5} = \frac{-2}{6} = -\frac{1}{3} \]

### Step 3: Write the equation of diagonal BD
Using the point-slope form of the equation of a line:
\[ y - y1 = m(x - x1) \]
Substituting \(m = -\frac{1}{3}\) and point B(-5, 6):
\[ y - 6 = -\frac{1}{3}(x + 5) \]
Expanding this:
\[ y - 6 = -\frac{1}{3}x - \frac{5}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 3y - 18 = -x - 5 \]
Rearranging gives:
\[ x + 3y = 13 \]
Thus, the equation of diagonal BD is:
\[ x + 3y = 13 \]

### Step 4: Find the slope of diagonal AC
Since diagonals of a rhombus bisect each other at right angles, the slope of diagonal AC (m_{AC}) is the negative reciprocal of the slope of BD:
\[ m_{AC} = -\frac{1}{m_{BD}} = -\frac{1}{-\frac{1}{3}} = 3 \]

### Step 5: Write the equation of diagonal AC
Using the midpoint O(-2, 5) and the slope m_{AC} = 3:
\[ y - 5 = 3(x + 2) \]
Expanding this:
\[ y - 5 = 3x + 6 \]
Rearranging gives:
\[ y = 3x + 11 \]
Thus, the equation of diagonal AC is:
\[ y = 3x + 11 \]

### Summary of Results
- The equation of diagonal BD is \(x + 3y = 13\).
- The equation of diagonal AC is \(y = 3x + 11\).