Let's address each part of the problem using the given points $B(-2, 3)$ and $C(-2, -4)$.

(i)

(a) Write the coordinates of $A$.

Since $BC$ is parallel to the y-axis, the x-coordinate of both points $B$ and $C$ is the same.

The points $D$ and $E$ are where lines $AB$ and $AC$ intersect the x-axis. The x-intercepts are crucial to find the equation of the lines $AB$ and $AC$, but we don't have enough information to calculate the coordinates of $A$ directly. However, in many problems like this, $A$ might be the point of intersection of lines $AB$ and $AC$ or might be given directly or implied by the geometry. Let's proceed under the assumption we need more information or a specific condition about $A$.

If any additional information about $A$ is provided, such as its location or any other relation, we can determine its coordinates directly.

(b) Find the lengths of $AB$ and $AC$.

The lengths of the line segments $AB$ and $AC$ can be found using the distance formula:

$\text{Length of } AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$ $\text{Length of } AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2}$

Given that $B(-2, 3)$ and $C(-2, -4)$, we need $A(x_A, y_A)$ to find the lengths. Without the coordinates of $A$, we can't find these lengths directly.

(c) Find the ratio in which $C$ divides $AB$.

To find the ratio in which $C$ divides $AB$, we can use the section formula, but we again need the coordinates of $A$. The section formula states:

$\text{If } C \text{ divides } AB \text{ in the ratio } m:n, \text{ then } C = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$

Given $B(-2, 3)$ and $C(-2, -4)$, and if we had $A$'s coordinates, we could solve for the ratio $m:n$.

(d) Find the equation of the line $BC$.

The equation of the line passing through points $B(-2, 3)$ and $C(-2, -4)$ can be found using the point-slope form of the line equation. Since the x-coordinate is constant (both points have $x = -2$), the line $BC$ is vertical.

For a vertical line, the equation is simply:

$x = -2$

This is the equation of line $BC$.

---

Summary:

• (a) The coordinates of $A$ are not provided or directly solvable without additional information.
• (b) The lengths $AB$ and $AC$ require the coordinates of $A$.
• (c) The ratio in which $C$ divides $AB$ also requires the coordinates of $A$.
• (d) The equation of the line $BC$ is $x = -2$.

Do you have any further details about point $A$, or would you like to clarify any other parts?

Follow-up Questions:

1. Do you have the coordinates for point $A$?
2. Are there any additional conditions that might help determine the position of point $A$?
3. Would you like to understand how to calculate lengths using the distance formula in more detail?
4. Do you want to explore different methods to find the ratio in which a point divides a line segment?
5. Should I explain more about vertical lines and their equations?
6. Would you like to discuss how to handle such problems when information is incomplete?
7. Do you need assistance with determining points of intersection between lines?
8. Would you like to learn more about the section formula and its applications?

Tip:

In coordinate geometry problems, carefully check the given conditions and the problem's constraints. Identifying missing information early can save time and help in making logical assumptions.