2. The points $$ A(2,3), B(1,8), C(6,11) $$ and $$ D(3,6) $$ are the vertices of a quadrilateral. a. Find the gradients of $$ A B $$ and $$ D C $$. What does this tell you about the lines $$ A B $$ and $$ D C $$ ? b. Find the equation of the diagonal joining $$ B $$ to $$ D $$. c. The diagonal $$ A C $$ has the equation $$ y=x+5 $$. Find the point of intersection of the two diagonals $$ A C $$ and $$ B D $$. d. Find the midpoint of $$ A C $$ and compare this point with the point of intersection of the diagonals $$ A C $$ and $$ B D $$.

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**Problem 2.b: Find the equation of the diagonal joining $$ B $$ to $$ D $$.**

To find the equation of the line $$ BD $$ connecting points $$ B(1,8) $$ and $$ D(3,6) $$, follow these steps:

1. **Calculate the Gradient (Slope) of $$ BD $$:**
   
   The gradient $$ m $$ of a line passing through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$
   
   Applying this to points $$ B(1,8) $$ and $$ D(3,6) $$:
   $$
   m = \frac{6 - 8}{3 - 1} = \frac{-2}{2} = -1
   $$

2. **Use the Point-Slope Form to Find the Equation:**
   
   The point-slope form of a line is:
   $$
   y - y_1 = m(x - x_1)
   $$
   
   Using point $$ B(1,8) $$ and $$ m = -1 $$:
   $$
   y - 8 = -1(x - 1)
   $$
   
   Simplifying:
   $$
   y - 8 = -x + 1
   $$
   $$
   y = -x + 9
   $$

Alternatively, the equation can be written in slope-intercept form:
   $$
   y = -x + 9
   $$

**Answer:**  
After simplifying, the diagonal BD has slope –1 and equation y = –x + 9.
The equation of the diagonal $$ BD $$ is $$ y = -x + 9 $$.

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