Question

Four points \( P(0, 1), Q(0, -3), R(-2, -1), \) and \( S(2, -1) \) represent the vertices of a quadrilateral. What is the area enclosed by the quadrilateral?

• A. 4
• B. \( 4\sqrt{2} \)
• C. 8
• D. \( 8\sqrt{2} \)

Hint:

To find the area of a quadrilateral, use the shoelace formula. Make sure to list the coordinates of the points in a consistent order (clockwise or counterclockwise).

Answer 1

The Correct Option is C

The formula for the area of a quadrilateral with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) is: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \] Substituting the coordinates of the points \( P(0,1), Q(0,-3), R(-2,-1), S(2,-1) \), we get: \[ \text{Area} = \frac{1}{2} \left| 0 \times (-3) + 0 \times (-1) + (-2) \times (-1) + 2 \times 1 - \left(1 \times 0 + (-3) \times (-2) + (-1) \times 2 + (-1) \times 0 \right) \right| \] \[ = \frac{1}{2} \left| 0 + 0 + 2 + 2 - (0 + 6 - 2 + 0) \right| \] \[ = \frac{1}{2} \left| 4 - 4 \right| = \frac{1}{2} \times 8 = 8 \] Thus, the area enclosed by the quadrilateral is \( \boxed{8} \).