Suppose ABCD is a quadrilateral such that the coordinates of A, B and C are `(1,3)(-2,6) and (5,-8)` respectively. For what choices of coordinates of D will make ABCD a trapezium ?

Options

A. (3, -6)
B. (6, -9)
C. (0, 5)
D. (3, -1)

Answer

To determine the coordinates of point D that will make quadrilateral ABCD a trapezium, we need to find conditions under which one pair of opposite sides is parallel. In this case, we will consider the sides AB and CD to be parallel.

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let the coordinates of points A, B, and C be:
- A(1, 3)
- B(-2, 6)
- C(5, -8)
- Let the coordinates of point D be (h, k).

2. **Calculate the Slope of Line AB:**
- The slope of a line through two points (x1, y1) and (x2, y2) is given by:
\[ \text{slope} = \frac{y2 - y1}{x2 - x1} \]
- For line AB:
\[ \text{slope of AB} = \frac{6 - 3}{-2 - 1} = \frac{3}{-3} = -1 \]

3. **Calculate the Slope of Line CD:**
- The slope of line CD will be:
\[ \text{slope of CD} = \frac{k - (-8)}{h - 5} = \frac{k + 8}{h - 5} \]

4. **Set the Slopes Equal for Parallel Lines:**
- For ABCD to be a trapezium with AB parallel to CD, we set the slopes equal:
\[ -1 = \frac{k + 8}{h - 5} \]

5. **Cross Multiply to Solve for k:**
- Cross multiplying gives:
\[ -1(h - 5) = k + 8 \]
\[ -h + 5 = k + 8 \]
\[ k = -h + 5 - 8 \]
\[ k = -h - 3 \]

6. **Conclusion:**
- The coordinates of point D must satisfy the equation:
\[ k = -h - 3 \]
- Thus, any point D(h, k) where \( k = -h - 3 \) will make ABCD a trapezium.