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

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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.
To find the perimeter of a rectangle using vertex coordinates: 1. **Label the Vertices:** Assign coordinates to each corner. 2. **Identify Adjacent Sides:** Determine which sides are next to each other. 3. **Calculate Side Lengths:** Use the distance formula for each side. 4. **Sum the Lengths:** Add up all four side lengths. 5. **Perimeter Formula:** Multiply the sum of two adjacent sides by 2. This process gives the rectangle's perimeter based on its vertex coordinates.

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

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