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 ensure that one pair of opposite sides is parallel. In this case, we will check if sides AB and DC can be parallel.

### Step-by-Step Solution:

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

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

3. **Set Up the Slope of DC**:
The slope of line segment DC (from D(h, k) to C(5, -8)) should also be -1 for AB and DC to be parallel:
\[ \text{slope of DC} = \frac{-8 - k}{5 - h} \]
Setting the slopes equal gives:
\[ \frac{-8 - k}{5 - h} = -1 \]

4. **Cross Multiply and Simplify**:
Cross multiplying gives:
\[ -8 - k = -1(5 - h) \]
Simplifying this:
\[ -8 - k = -5 + h \]
Rearranging gives:
\[ h + k + 3 = 0 \]

5. **Finding Possible Coordinates for D**:
The equation \( h + k + 3 = 0 \) can be rearranged to find k in terms of h:
\[ k = -h - 3 \]
This means that for any value of h, we can find a corresponding k that will satisfy the equation.

6. **Check Given Options**:
We need to check which of the given options satisfy this equation:
- Option 1: (3, -6) → \( 3 + (-6) + 3 = 0 \) (Valid)
- Option 2: (6, -9) → \( 6 + (-9) + 3 = 0 \) (Valid)
- Option 3: (0, 5) → \( 0 + 5 + 3 = 8 \) (Invalid)
- Option 4: (3, -1) → \( 3 + (-1) + 3 = 5 \) (Invalid)

### Conclusion:
The coordinates of D that will make ABCD a trapezium are:
- (3, -6)
- (6, -9)