What is the area of this quadrilateral?  Points at A:1.5,-4 B:5,1 C:-1.5,5 D:-5,0

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.

Learn how to calculate the area of a quadrilateral using the Shoelace formula with detailed step-by-step instructions. This problem involves finding the area of a quadrilateral given vertices at (1.5, -4), (5, 1), (-1.5, 5), and (-5, 0). Suitable for advanced high school students and those studying coordinate geometry.

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