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### (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

Diagram 6 shows a rectangle ABCD on a Cartesian plane. (a) Show that y = x + 12. (b) Find the area of ΔABC in terms of x and y. (c) Hence, find the coordinates of A given that the area of the rectangle ABCD is 120 units².

This problem involves proving a relationship between x and y in a rectangle on a Cartesian plane, calculating the area of triangle ABC, and finding the coordinates of point A. The solution covers key concepts in geometry and coordinate geometry suitable for students in grades 10-12.

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