The problem asks to find the coordinates of point $D$ such that line $CD$ is parallel to line $AB$. Also, it requests to show algebraically that the two lines are parallel.

Steps to solve the problem:

1. Determine the slope of line $AB$:

   • First, identify the coordinates of points $A$ and $B$.
   • The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$
   • Using the graph, let's extract the coordinates of $A$ and $B$.

2. Find the slope of line $CD$:

   • The line $CD$ must have the same slope as line $AB$ since the lines are parallel.
   • Identify the coordinates of $C$, and use the slope equation to determine the coordinates of $D$.

3. Solve algebraically to ensure the lines are parallel.

   • Set the slope of $AB$ equal to the slope of $CD$.
   • Solve for the coordinates of $D$.

I’ll begin by analyzing the points from the graph in the image to determine the specific coordinates and calculate the slopes.Let's extract the coordinates from the graph and work through the solution:

Step 1: Coordinates of $A$ and $B$

By observing the graph:

• $A$ appears to be at the point $(0, -2)$ based on the grid.
• $B$ appears to be at the point $(2, 1)$.

Step 2: Calculate the slope of line $AB$

Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ for points $A(0, -2)$ and $B(2, 1)$: $m_{AB} = \frac{1 - (-2)}{2 - 0} = \frac{1 + 2}{2} = \frac{3}{2}$ So, the slope of line $AB$ is $\frac{3}{2}$.

Step 3: Coordinates of $C$

Looking at the graph, point $C$ seems to be at $(-2, 0)$.

Step 4: Find the coordinates of $D$

To ensure that line $CD$ is parallel to line $AB$, the slope of $CD$ must also be $\frac{3}{2}$. Let the coordinates of $D$ be $(x_D, y_D)$. Using the slope formula for points $C(-2, 0)$ and $D(x_D, y_D)$: $m_{CD} = \frac{y_D - 0}{x_D - (-2)} = \frac{y_D}{x_D + 2}$ Set this equal to the slope of line $AB$, which is $\frac{3}{2}$: $\frac{y_D}{x_D + 2} = \frac{3}{2}$ Cross-multiply to solve for $y_D$: $2y_D = 3(x_D + 2)$ $2y_D = 3x_D + 6$ $y_D = \frac{3x_D + 6}{2}$

Thus, any point $D(x_D, y_D)$ on line $CD$ can be found by choosing a specific value for $x_D$ and solving for $y_D$. For instance, if $x_D = 0$ (a logical choice based on the graph's symmetry), we have: $y_D = \frac{3(0) + 6}{2} = \frac{6}{2} = 3$

So, the coordinates of $D$ could be $(0, 3)$.

Final Coordinates:

The coordinates of point $D$ such that line $CD$ is parallel to line $AB$ are $D(0, 3)$.

Would you like more details on any part of the solution, or any clarification?

Here are five related questions to explore further:

1. How do we determine if two lines are perpendicular instead of parallel?
2. What is the slope of a horizontal line, and how does it compare to a vertical line?
3. Can two lines with the same slope ever intersect?
4. What is the equation of line $AB$ in slope-intercept form?
5. How can you find the midpoint of line segment $AB$?

Tip: When two lines are parallel, their slopes are equal. Use this property to verify parallelism algebraically.