Let's go through each part of the question based on the image you've uploaded.

(a) Show that $y = x + 12$

We need to demonstrate that the coordinates of point $A(x, y)$ satisfy the equation $y = x + 12$.

• From the diagram, we can see that points $A(x, y)$, $B(-4, 8)$, $C(6, -2)$, and $D$ form a rectangle, and $O$ is the origin.
• Since $A(x, y)$ and $B(-4, 8)$ are on the same diagonal, and diagonals of a rectangle bisect each other at right angles, we can use the midpoint formula and the slope relationship to show that $y = x + 12$.

Let's now apply the midpoint formula and confirm the relationship:

• The midpoint of $AC$ and $BD$ is the origin, point $O(0, 0)$.
• The midpoint formula for $A(x, y)$ and $C(6, -2)$ is: $\left( \frac{x + 6}{2}, \frac{y - 2}{2} \right) = (0, 0)$ This gives two equations:
  1. $\frac{x + 6}{2} = 0$ → $x + 6 = 0$ → $x = -6$
  2. $\frac{y - 2}{2} = 0$ → $y - 2 = 0$ → $y = 2$

However, this suggests the coordinates of $A$ are incorrect in the context of our earlier steps. Therefore, reviewing this