11. COSS. Identify Structure Explain the steps you would use to find the perimeter of a rectangle using the coordinates of the vertices.

1. **Label the Vertices:** Suppose the rectangle's vertices are given as \[ A(x_A, y_A),\quad B(x_B, y_B),\quad C(x_C, y_C),\quad D(x_D, y_D). \] Ensure that the vertices are ordered consecutively (either clockwise or counterclockwise). If they are not, reorder them so that consecutive vertices are adjacent in the rectangle. 2. **Identify the Sides:** The sides of the rectangle are the segments \( \overline{AB} \), \( \overline{BC} \), \( \overline{CD} \), and \( \overline{DA} \). The rectangle will have two distinct side lengths (the length and the width). 3. **Calculate the Lengths Using the Distance Formula:** For any two adjacent vertices \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance (length of the side) is given by \[ \text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. \] - Compute \( \overline{AB} \) as \[ AB = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2}. \] - Compute \( \overline{BC} \) as \[ BC = \sqrt{(x_C-x_B)^2 + (y_C-y_B)^2}. \] - Since opposite sides of a rectangle are equal, \( CD = AB \) and \( DA = BC \). 4. **Determine the Perimeter:** The perimeter \( P \) of a rectangle is the sum of all its side lengths. Thus, \[ P = AB + BC + CD + DA. \] Because \( CD = AB \) and \( DA = BC \), this simplifies to \[ P = 2(AB + BC). \] 5. **Summarize the Process:** - Label and order the vertices correctly. - Use the distance formula to compute the lengths of two adjacent sides. - Multiply the sum of these two side lengths by 2 to obtain the perimeter. Following these steps will allow you to find the perimeter of a rectangle using the coordinates of its vertices.