https://file.aiquickdraw.com/mathbot/file/1729170763081gqfmyk9i.jpg

Let's go step by step and solve the questions based on the given image:

### a. Write the coordinates of H.

Looking at the graph, we can count the position of point \( H \). It is located at:
- **X-coordinate**: 3 (three units to the right of point \( P \))
- **Y-coordinate**: 5 (five units up from point \( P \))

Thus, the coordinates of \( H \) are:
\[
H(3, 5)
\]

### b. Find the distance of the point G from the Y-axis.

Point \( G \) is at:
\[
G(6, 4)
\]
To find the distance from the \( Y \)-axis, we take the absolute value of the \( X \)-coordinate of \( G \), which is 6. Therefore, the distance from the \( Y \)-axis is:
\[
\text{Distance} = |6| = 6 \, \text{units}
\]

### c. What will be the length of side HG?

The coordinates of \( H \) are \( H(3, 5) \) and the coordinates of \( G \) are \( G(6, 4) \). To find the distance between two points, we use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of \( H \) and \( G \):
\[
d = \sqrt{(6 - 3)^2 + (4 - 5)^2} = \sqrt{(3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
Thus, the length of side \( HG \) is:
\[
HG = \sqrt{10} \, \text{units}
\]

### d. Find the length of diagonal FD and the value of x.

The coordinates of \( D \) are \( D(2, 1) \) and the coordinates of \( F \) are \( F(x, 8) \). To find the length of diagonal \( FD \), we use the distance formula again:
\[
FD = \sqrt{(x - 2)^2 + (8 - 1)^2} = \sqrt{(x - 2)^2 + 7^2} = \sqrt{(x - 2)^2 + 49}
\]
This expression depends on the value of \( x \), so we need more information to calculate the exact value of \( FD \).

Do you want me to provide further clarification or solve any part in more detail?

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### Related Questions:
1. How do you derive the distance formula from two points?
2. What are the coordinates of point \( F \) when \( x = 5 \)?
3. Can you find the equation of the line passing through points \( H \) and \( G \)?
4. What is the slope of line \( FD \) for any value of \( x \)?
5. How would you calculate the area of the shape formed by points \( E \), \( A \), \( D \), and \( H \)?

**Tip**: When solving problems involving coordinates, it’s useful to sketch the graph and visualize the positions to avoid errors with calculations.

Write the coordinates of H. Find the distance of the 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.

Solve a coordinate geometry problem involving points H, G, and F. Calculate the coordinates of H, the distance from G to the Y-axis, the length of side HG, and diagonal FD. Apply the distance formula and Pythagorean theorem, suitable for students in Grades 7-10.

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