Find the area of the polygon with the vertices P(1,1), Q(-2,1), R(-1,-4)

To find the area of a polygon with given vertices, we can use the **Shoelace Theorem** (or **Gauss's area formula**). For a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\) in order, the area \( A \) is given by:

\[
A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \dots + x_{n-1}y_n + x_ny_1 - (y_1x_2 + y_2x_3 + \dots + y_{n-1}x_n + y_nx_1) \right|
\]

For the vertices \( P(1, 1) \), \( Q(-2, 1) \), and \( R(-1, -4) \), we can apply the formula as follows:

### Step 1: List the coordinates in order:
- \( P(1, 1) \)
- \( Q(-2, 1) \)
- \( R(-1, -4) \)
- Back to \( P(1, 1) \) to close the polygon.

### Step 2: Apply the Shoelace Theorem:

\[
A = \frac{1}{2} \left| (1 \cdot 1 + (-2) \cdot (-4) + (-1) \cdot 1) - (1 \cdot (-2) + 1 \cdot (-1) + (-4) \cdot 1) \right|
\]

### Step 3: Calculate each part:

\[
A = \frac{1}{2} \left| (1 + 8 - 1) - (-2 - 1 - 4) \right|
\]
\[
A = \frac{1}{2} \left| 8 - (-7) \right|
\]
\[
A = \frac{1}{2} \times 15 = \frac{15}{2} = 7.5
\]

### Final Answer:
The area of the triangle is \( \boxed{7.5} \) square units.

Would you like more details on this process, or do you have any questions?

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Here are some related questions you might find interesting:

1. How does the Shoelace Theorem extend to polygons with more vertices?
2. Can the area formula be used for concave polygons?
3. What is the difference between the Shoelace Theorem and Heron's formula for triangle area?
4. How would the area change if one of the vertices was moved?
5. Can the Shoelace Theorem be derived from vector cross products?

**Tip:** Always ensure the vertices are listed in a consistent order (clockwise or counterclockwise) when using the Shoelace Theorem to avoid incorrect results.

Learn how to find the area of a polygon with vertices P(1,1), Q(-2,1), R(-1,-4) using the Shoelace Theorem (Gauss's area formula). Follow a step-by-step solution and understand the calculation process. Suitable for high school students and anyone interested in geometry.

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