Questions: Find the area of the polygon with the given vertices. N(-4,1), P(1,1), Q(3,-1), R(-2,-1) The area is square units.

Solution Steps

To find the area of a polygon given its vertices, we can use the Shoelace formula (also known as Gauss's area formula). This formula is particularly useful for polygons with vertices defined by their coordinates. The formula is:

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

Where \((x_i, y_i)\) are the coordinates of the vertices of the polygon.

1. List the coordinates of the vertices in order.
2. Apply the Shoelace formula to calculate the area.
3. Implement the formula in Python to compute the area.

Given vertices of the polygon are: \[ N(-4, 1), P(1, 1), Q(3, -1), R(-2, -1) \]

The Shoelace formula for the area of a polygon with vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is: \[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \]

Substitute the given vertices into the formula: \[ \begin{align_} \text{Area} &= \frac{1}{2} \left| (-4 \cdot 1 + 1 \cdot (-1) + 3 \cdot (-1) + (-2) \cdot 1) - (1 \cdot 1 + 1 \cdot 3 + (-1) \cdot (-2) + (-1) \cdot (-4)) \right| \\ &= \frac{1}{2} \left| (-4 - 1 - 3 - 2) - (1 + 3 + 2 + 4) \right| \\ &= \frac{1}{2} \left| -10 - 10 \right| \\ &= \frac{1}{2} \left| -20 \right| \\ &= \frac{1}{2} \times 20 \\ &= 10 \end{align_} \]

Final Answer