https://file.aiquickdraw.com/mathbot/file/1729171906602jatuyg57.jpg

Let's analyze the image and solve the questions based on the given diagram.

### Information from the Image:
- The diagram is a grid with points labeled such as \( H \), \( G \), \( F \), \( D \), \( C \), \( B \), \( A \), and \( E \).
- \( P \) is considered the origin (bottom-left corner), and points are defined in a grid of squares, each square likely representing 1 unit.

### Questions:

**a. Write the coordinates of H.**
- From the diagram, \( H \) is located at (2, 3) on the grid.

**b. Find the distance of the point G from the Y-axis.**
- The x-coordinate of point \( G \) is 5, so the distance from the Y-axis (which is at \( x = 0 \)) is 5 units.

**c. What will be the length of side HG?**
- \( H = (2, 3) \), \( G = (5, 3) \)
- Since both points have the same y-coordinate, the length of side \( HG \) is the horizontal distance, which is:
  \[
  HG = |5 - 2| = 3 \text{ units}
  \]

**d. Find the length of diagonal FD and the value of x.**
- \( F \) is given as \( (x, 8) \), and \( D = (2, 2) \).
- To find the length of diagonal \( FD \), we use the distance formula:
  \[
  \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  \]
  Substituting the values:
  \[
  FD = \sqrt{(x - 2)^2 + (8 - 2)^2} = \sqrt{(x - 2)^2 + 36}
  \]
  Now, the length of the diagonal \( FD \) is not provided, but once the value of \( x \) is known, we can calculate this length.

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### Would you like to clarify or go into more detail on any step?

Here are 5 related questions for further exploration:
1. What would be the length of diagonal \( HB \)?
2. How would you calculate the area of quadrilateral \( HGDC \)?
3. If the coordinates of point \( A \) are needed, how can you find them from the diagram?
4. What is the total perimeter of the shape formed by points \( H, G, F, D \)?
5. Can you calculate the slope of line \( HG \)?

### Tip:
Always use the distance formula or grid counting to find the lengths between points in coordinate geometry.

Write the coordinates of H. Find the distance of point G from the Y-axis. What will be the length of side HG? Find the length of diagonal FD and the value of x.

Learn how to solve coordinate geometry problems including finding the coordinates of points, distance from axes, and lengths of diagonals using the Pythagorean theorem and distance formula. Suitable for students in grades 6-8.

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