Identify the area of the polygon with vertices \( P(1,2) \), \( Q(1,4), R(-1,6) \), and \( S(-3,2) \).

To find the area of the polygon with vertices \( P(1,2) \), \( Q(1,4) \), \( R(-1,6) \), and \( S(-3,2) \), we can use the Shoelace formula. The Shoelace formula is a method for calculating the area of a simple polygon whose vertices are given by their coordinates in the plane. The Shoelace formula is given by: \[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right| \] where \( (x_i, y_i) \) are the coordinates of the vertices of the polygon, and \( n \) is the number of vertices. Let's calculate the area of the polygon using the Shoelace formula. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|\left(1\times 4-1\times 2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step1: Calculate: \(\frac{1}{2}\left|\left(4-1\times 2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step2: Multiply the numbers: \(\frac{1}{2}\left|\left(4-2\right)+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step3: Subtract the numbers: \(\frac{1}{2}\left|2+\left(1\times 6-\left(-1\right)\times 4\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step4: Remove the parentheses: \(\frac{1}{2}\left|2+\left(1\times 6-\left(-4\right)\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step5: Calculate: \(\frac{1}{2}\left|2+\left(6-\left(-4\right)\right)+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step6: Subtract the terms: \(\frac{1}{2}\left|2+10+\left(\left(-1\right)\times 2-\left(-3\right)\times 6\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step7: Remove the parentheses: \(\frac{1}{2}\left|2+10+\left(-2-\left(-3\times 6\right)\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step8: Multiply the numbers: \(\frac{1}{2}\left|2+10+\left(-2-\left(-18\right)\right)+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step9: Subtract the terms: \(\frac{1}{2}\left|2+10+16+\left(\left(-3\right)\times 2-1\times 6\right)\right|\) - step10: Remove the parentheses: \(\frac{1}{2}\left|2+10+16+\left(-3\times 2-1\times 6\right)\right|\) - step11: Multiply the numbers: \(\frac{1}{2}\left|2+10+16+\left(-6-1\times 6\right)\right|\) - step12: Multiply the numbers: \(\frac{1}{2}\left|2+10+16+\left(-6-6\right)\right|\) - step13: Subtract the numbers: \(\frac{1}{2}\left|2+10+16+\left(-12\right)\right|\) - step14: Remove the parentheses: \(\frac{1}{2}\left|2+10+16-12\right|\) - step15: Calculate: \(\frac{1}{2}\left|16\right|\) - step16: Calculate the absolute value: \(\frac{1}{2}\times 16\) - step17: Reduce the numbers: \(1\times 8\) - step18: Simplify: \(8\)