Let **ABCD** be a parallelogram such that the coordinates of its three vertices **A, B, C** are **(1, 1)**, **(3, 4)**, and **(-2, 8)**, respectively. Then, the coordinates of the vertex **D** are:

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To find the coordinates of vertex **D**, we use the property of a parallelogram that opposite sides are parallel and have equal slopes.

1. **Given vertices:**  
   A = (1, 1), B = (3, 4), C = (-2, 8).

2. **Slope of AB:**  
   $$
   	ext{Slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{3 - 1} = \frac{3}{2}.
   $$

Since ABCD is a parallelogram, the slope of DC must also be $$\frac{3}{2}$$.

3. **Possible coordinates for D:**  
   Using the options provided and ensuring that D lies to the left and below C (as shown in the diagram), we test (-4, 5) and (-3, 4).

4. **Verify slope of DC:**  
   For D = (-4, 5):  
   $$
   	ext{Slope of DC} = \frac{8 - 5}{-2 - (-4)} = \frac{3}{2}.
   $$  
   This matches the slope of AB.

For D = (-3, 4):  
   $$
   	ext{Slope of DC} = \frac{8 - 4}{-2 - (-3)} = \frac{4}{1} = 4.
   $$  
   This does not match the slope of AB.

5. **Conclusion:**  
   The coordinates of vertex D are **(-4, 5)**.

Alternatively: Observe that point C is shifted left by 5 units and up by 4 units from point B. Similarly, point D must be shifted left by 5 units and up by 4 units from point A:  
$$
D = (1 - 5, 1 + 4) = (-4, 5).

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