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### Given:
- The trapezium \( ABCD \), where \( AD \parallel BC \), \( AB \perp BC \), and point \( C \) lies on the x-axis.
- Coordinates of \( A(-18, 0) \) and \( B(0, 24) \).
- Equation of line \( CD \): \( x - 7y + k = 0 \).

### (a) Find the value of \( k \)
The point \( C(0, 24) \) lies on the line \( CD \). To find \( k \), substitute the coordinates of \( C \) into the line equation \( x - 7y + k = 0 \).

\[
0 - 7(24) + k = 0
\]
\[
k = 168
\]

So, \( k = 168 \).

---

### (b)(i) Find the equation of \( AD \)
Since \( AD \parallel BC \), and we know the slope of \( BC \) (from points \( B(0, 24) \) and \( C(0, 24) \)) is zero, \( AD \) must also have a slope of 0 (since parallel lines have equal slopes).

Therefore, the equation of \( AD \) is a horizontal line passing through \( A(-18, 0) \), which gives:
\[
y = 0
\]

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### (b)(ii) Find the area of trapezium \( ABCD \)
The area \( A \) of a trapezium is given by:
\[
A = \frac{1}{2} \times (a + b) \times h
\]
where \( a \) and \( b \) are the lengths of the parallel sides, and \( h \) is the height.

- The length of \( AD \) is 18 (since \( A(-18, 0) \)).
- The length of \( BC \) is also 18 (since \( B(0, 24) \) to \( C(0, 24) \)).
- The height is the vertical distance between \( AD \) and \( BC \), which is 24 (the y-coordinate of point \( B \)).

Thus, the area of the trapezium is:
\[
A = \frac{1}{2} \times (18 + 18) \times 24 = \frac{1}{2} \times 36 \times 24 = 432 \, \text{square units}
\]

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Let me know if you'd like more details on any step. Here are 5 related questions for you:

1. How do we derive the equation of a line given two points?
2. What is the geometric significance of parallel lines in a trapezium?
3. Why does the slope of a line determine its equation?
4. How does the height of a trapezium relate to its area?
5. Can you explain how to determine the slope of perpendicular lines?

**Tip**: Always check whether lines are parallel or perpendicular, as this can simplify finding equations!

ABCD is a trapezium. It is given that AD || BC, AB ⊥ BC, and C lies on the x-axis. The coordinates of A and B are (-18, 0) and (0, 24), respectively. The equation of CD is x - 7y + k = 0.
(a) Find the value of k.
(b)(i) Find the equation of AD.
(b)(ii) Hence, find the area of the trapezium ABCD.

Solve a coordinate geometry problem involving a trapezium ABCD, where AD is parallel to BC and AB is perpendicular to BC. Find the value of k for line CD, determine the equation of AD, and calculate the area of the trapezium using key geometric concepts and formulas.

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