Let's break down the question based on the diagram and details given:

Question Overview:

• You are dealing with a quadrilateral $ABCD$ where point $A$ lies on the $y$-axis, and we are given some coordinates and line equations.
• The equation of line $BC$ is $y - 2x + 9 = 0$.
• The goal is to find:
  1. The equation of the straight line $CD$ (part (a)(i)).
  2. The coordinates of point $C$ (part (a)(ii)).
  3. The area of quadrilateral $ABCD$ (part (b)).

Step-by-Step Solution:

Part (a)(i): Finding the Equation of Line $CD$

• Given Coordinates:

  • Point $D = (6, -1)$
  • We need to find the equation of line $CD$.

  The slope $m$ of line $CD$ can be found using the coordinates of points $C(x_1, y_1)$ and $D(6, -1)$. Once we find the coordinates of $C$ (from part (a)(ii)), we can use the formula for the slope of a line:

  $m_{CD} = \frac{y_2 - y_1}{x_2 - x_1}$

  Then, we can apply the point-slope form of a line equation:

  $y - y_1 = m(x - x_1)$

Part (a)(ii): Finding the Coordinates of Point $C$

• Given Information:

  • The equation of line $BC$ is $y - 2x + 9 = 0$.

  Solving for $y$:

  $y = 2x - 9$

  Point $C$ lies on the $x$-axis, meaning $y = 0$ for point $C$. Substituting this into the equation of the line $BC$:

  $0 = 2x - 9$

  Solving for $x$:

  $x = \frac{9}{2} = 4.5$

  So, the coordinates of point $C$ are $(4.5, 0)$.

Part (b): Finding the Area of Quadrilateral $ABCD$

• To find the area of quadrilateral $ABCD$, we can use the formula for the area of a polygon with known vertices:

  If the vertices are $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$, the area is given by:

  $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_4) + x_2(y_3 - y_1) + x_3(y_4 - y_2) + x_4(y_1 - y_3) \right|$

  Substituting the coordinates of $A(0, 5), B(x, 5), C(4.5, 0), D(6, -1)$, we can calculate the area.

Would you like to go through the final calculations and complete the solution?