Compute the perimeter and area of the quadrilateral determined by the points \( A(1,-2), B(5,-2) \), \( C(5,3) \), and \( D(1,3) \). \( P=\square \) unit \( A=\square \) unit \( ^{2} \) Question Help: Video Submit Question

**Step 1. Calculate the lengths of the sides** - For side \( AB \) (from \( A(1,-2) \) to \( B(5,-2) \)): \[ AB = \sqrt{(5-1)^2 + (-2 - (-2))^2} = \sqrt{4^2 + 0^2} = 4 \] - For side \( BC \) (from \( B(5,-2) \) to \( C(5,3) \)): \[ BC = \sqrt{(5-5)^2 + (3 - (-2))^2} = \sqrt{0^2 + 5^2} = 5 \] - For side \( CD \) (from \( C(5,3) \) to \( D(1,3) \)): \[ CD = \sqrt{(1-5)^2 + (3-3)^2} = \sqrt{(-4)^2 + 0^2} = 4 \] - For side \( DA \) (from \( D(1,3) \) to \( A(1,-2) \)): \[ DA = \sqrt{(1-1)^2 + (-2-3)^2} = \sqrt{0^2 + (-5)^2} = 5 \] **Step 2. Compute the Perimeter** Sum the lengths of all sides: \[ P = AB + BC + CD + DA = 4 + 5 + 4 + 5 = 18 \quad \text{units} \] **Step 3. Compute the Area** Since the quadrilateral is a rectangle, the area is the product of its length and width. The horizontal distance (length) is: \[ 5 - 1 = 4 \quad \text{units} \] and the vertical distance (width) is: \[ 3 - (-2) = 5 \quad \text{units} \] Thus, the area is: \[ A = 4 \times 5 = 20 \quad \text{square units} \] **Final Answer:** \(P = 18\) unit and \(A = 20\) unit\(^2\).