ABCD is a parallelogram where A (x, y), B (5, 8), C (4, 7) and D 2, -4). Find
(ii) Equation of diagonal BD.

Answer

To find the equation of diagonal BD in the parallelogram ABCD, we will use the coordinates of points B and D.

Given:
- B(5, 8)
- D(2, -4)

### Step 1: Identify the coordinates of points B and D.
We have:
- B = (5, 8)
- D = (2, -4)

### Step 2: Use the two-point form of the equation of a line.
The two-point form of the equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

### Step 3: Substitute the coordinates of points B and D into the formula.
Here, we have:
- (x1, y1) = (5, 8)
- (x2, y2) = (2, -4)

Substituting these values into the two-point form:
\[ y - 8 = \frac{-4 - 8}{2 - 5}(x - 5) \]

### Step 4: Simplify the equation.
Calculating the slope:
\[ \frac{-4 - 8}{2 - 5} = \frac{-12}{-3} = 4 \]

So, the equation becomes:
\[ y - 8 = 4(x - 5) \]

### Step 5: Expand and rearrange the equation.
Expanding the right-hand side:
\[ y - 8 = 4x - 20 \]

Now, adding 8 to both sides:
\[ y = 4x - 20 + 8 \]
\[ y = 4x - 12 \]

### Step 6: Write the final equation in standard form.
Rearranging gives:
\[ 4x - y - 12 = 0 \]

Thus, the equation of diagonal BD is:
\[ 4x - y = 12 \]