Determine the area of the quadrilateral whose vertices are (–10, –10), (4, –24), (10, 10) and (–14, -2).

To determine the area of a quadrilateral given its vertices, we can use the **Shoelace Theorem** (or Gauss's area formula). The vertices of the quadrilateral are given as \((-10, -10)\), \((4, -24)\), \((10, 10)\), and \((-14, -2)\).

### Step 1: List the coordinates
We list the coordinates in order, and repeat the first vertex at the end to close the loop:
\[
\begin{array}{cc}
x & y \\
-10 & -10 \\
4 & -24 \\
10 & 10 \\
-14 & -2 \\
-10 & -10 \\
\end{array}
\]

### Step 2: Apply the Shoelace Theorem
According to the Shoelace Theorem, the area \(A\) is given by:
\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1}) - \sum_{i=1}^{n-1} (y_i x_{i+1}) \right|
\]
Substituting the values:

\[
\text{Sum of } x_i y_{i+1} = (-10 \times -24) + (4 \times 10) + (10 \times -2) + (-14 \times -10)
\]
\[
\text{Sum of } y_i x_{i+1} = (-10 \times 4) + (-24 \times 10) + (10 \times -14) + (-2 \times -10)
\]

### Step 3: Calculate the determinants
Let's compute each step:

\[
(-10 \times -24) + (4 \times 10) + (10 \times -2) + (-14 \times -10) = 240 + 40 - 20 + 140 = 400
\]

\[
(-10 \times 4) + (-24 \times 10) + (10 \times -14) + (-2 \times -10) = -40 - 240 - 140 + 20 = -400
\]

### Step 4: Calculate the area
Now, subtract and take the absolute value:

\[
A = \frac{1}{2} \left| 400 - (-400) \right| = \frac{1}{2} \times 800 = 400
\]

So, the area of the quadrilateral is **400 square units**.

Would you like further details or have any questions?

### Related Questions
1. How does the Shoelace Theorem work with polygons having more than four vertices?
2. Can you derive the Shoelace Theorem formula from scratch?
3. What would the area be if the vertices were arranged differently?
4. How do you determine the orientation of a quadrilateral?
5. Can the Shoelace Theorem be applied to a triangle? How?

**Tip:** Ensure to list vertices in the correct sequence (clockwise or counterclockwise) to get the correct area calculation using the Shoelace Theorem.

Learn how to calculate the area of a quadrilateral using the Shoelace Theorem with a step-by-step solution. This problem involves applying coordinate geometry to find the area of a quadrilateral given its vertices. Suitable for high school students.

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