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?

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.