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The problem in the image asks to determine the coordinates of points \( A \) and \( B \), with the following details:

- The line \( g_2 \) has the equation \( y = x + 2 \).
- Lines \( g_2 \) and \( g_3 \) are parallel.
- We also observe the line \( g_1 \), which appears to be the horizontal axis or another given line.

### Step-by-step approach:

#### 1. **Equation of \( g_2 \)**:
   The equation of \( g_2 \) is given as \( y = x + 2 \), which is a straight line with a slope of 1 and a y-intercept at \( y = 2 \).

#### 2. **Parallel Lines \( g_2 \) and \( g_3 \)**:
   Since \( g_2 \) and \( g_3 \) are parallel, they must have the same slope. Therefore, the equation of \( g_3 \) will also have a slope of 1 but a different y-intercept. Let’s assume the equation of \( g_3 \) is of the form \( y = x + c \), where \( c \) is the unknown y-intercept.

   From the diagram, we can estimate that \( g_3 \) passes through the point (0, 12). Hence, the equation of \( g_3 \) becomes:
   \[
   g_3: y = x + 12
   \]

#### 3. **Finding Point \( A \)**:
   The point \( A \) lies on the line \( g_3 \) and has a y-coordinate of 7 (as given by the diagram). So, we substitute \( y = 7 \) into the equation of \( g_3 \) to find the corresponding x-coordinate of \( A \):

   \[
   7 = x + 12
   \]
   Solving for \( x \):
   \[
   x = 7 - 12 = -5
   \]
   Thus, the coordinates of point \( A \) are \( (-5, 7) \).

#### 4. **Finding Point \( B \)**:
   The point \( B \) lies on the line \( g_2 \) and has a y-coordinate of 8 (from the diagram). We substitute \( y = 8 \) into the equation of \( g_2 \) to find the corresponding x-coordinate of \( B \):

   \[
   8 = x + 2
   \]
   Solving for \( x \):
   \[
   x = 8 - 2 = 6
   \]
   Therefore, the coordinates of point \( B \) are \( (6, 8) \).

### Final Answer:
- \( A(-5, 7) \)
- \( B(6, 8) \)

Would you like more details or have any other questions? Here are 5 related questions to expand your understanding:

1. What is the significance of parallel lines having the same slope?
2. How can you find the equation of a line given two points?
3. How would the solution change if the lines were perpendicular instead of parallel?
4. Can you derive the slope-intercept form from two points on a line?
5. How can you determine the distance between two points on a graph?

**Tip**: When solving geometry problems involving parallel lines, always check if the slopes are equal to ensure the lines don't intersect.

Determine the coordinates of points A and B given the parallel lines g2: y = x + 2 and g3, with additional data from the diagram.

In this problem, we calculate the coordinates of points A and B using the equations of parallel lines g2: y = x + 2 and g3. Learn step-by-step how to apply the properties of parallel lines and solve using linear equations. Suitable for students in Grade 8-10.

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