To find the area of a quadrilateral given its vertices, we can use the Shoelace formula (also known as Gauss's area formula). This formula is suitable for any simple polygon.

Given the points: $A(1.5, -4)$ $B(5, 1)$ $C(-1.5, 5)$ $D(-5, 0)$

The Shoelace formula for the area $A$ of a quadrilateral with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$ is:

$A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Substituting the given points:

$x_1 = 1.5, \; y_1 = -4$ $x_2 = 5, \; y_2 = 1$ $x_3 = -1.5, \; y_3 = 5$ $x_4 = -5, \; y_4 = 0$

Now, calculate the terms:

$A = \frac{1}{2} \left| (1.5 \cdot 1) + (5 \cdot 5) + (-1.5 \cdot 0) + (-5 \cdot -4) - ((-4 \cdot 5) + (1 \cdot -1.5) + (5 \cdot -5) + (0 \cdot 1.5)) \right|$

$A = \frac{1}{2} \left| 1.5 + 25 + 0 + 20 - (-20 - 1.5 - 25 + 0) \right|$

$A = \frac{1}{2} \left| 46.5 - (-46.5) \right|$

$A = \frac{1}{2} \left| 46.5 + 46.5 \right|$

$A = \frac{1}{2} \left| 93 \right|$

$A = \frac{1}{2} \cdot 93$

$A = 46.5$

The area of the quadrilateral is $46.5$ square units.

Would you like more details or have any questions?

Here are 5 more questions you might ask:

1. How do you find the area of a triangle using its vertices?
2. What is the Shoelace theorem?
3. Can you explain the steps of using the Shoelace formula in more detail?
4. How do you determine if a set of points forms a convex or concave polygon?
5. Can you help with finding the centroid of a polygon?

Tip: Always double-check your coordinates and calculations when using the Shoelace formula to ensure accuracy.