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

To find the area of a polygon, we can use the **shoelace formula** (or **Gauss's area formula**) for polygons, which is effective when the vertices of the polygon are given.

For a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\), the area \(A\) is given by:

\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|
\]

For the triangle with vertices \(P(1,1)\), \(Q(-2,1)\), and \(R(-1,-4)\), let's apply the formula:

1. \(P(1, 1)\)
2. \(Q(-2, 1)\)
3. \(R(-1, -4)\)

Now apply the coordinates to the formula:

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

Let's compute that step by step.

Learn how to calculate the area of a triangle with vertices P(1,1), Q(-2,1), and R(-1,-4) using the shoelace formula. This problem is suitable for students in Grades 9-12, covering key geometry concepts and the application of coordinate geometry in polygon area calculations.

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