I can help you with the math problem in the image you've uploaded. It seems to contain three tasks related to a rectangular plot of land with trees and a smaller inner rectangle labeled $PQRS$.

Let me summarize and solve each part:

Part (a) - Find the width of rectangle $PQRS$

From the diagram, it appears there is a coordinate grid with points marked at each meter along the boundary. The distance between any two adjacent points on the boundary is 1 meter.

Let’s identify the coordinates of $P, Q, R,$ and $S$:

• Point $P$ appears to be at $(2, 8)$
• Point $Q$ appears to be at $(2, 4)$
• Point $R$ appears to be at $(6, 4)$
• Point $S$ appears to be at $(6, 8)$

The width of the rectangle $PQRS$ can be found by calculating the horizontal or vertical distance between the points. The horizontal distance between $P$ and $S$ (or $Q$ and $R$) is $6 - 2 = 4$ meters, and the vertical distance between $P$ and $Q$ (or $S$ and $R$) is $8 - 4 = 4$ meters.

Thus, the width of the rectangle $PQRS$ is 4 meters.

Part (b) - Find the distance between the vertices of diagonal $QS$

To find the distance between points $Q(2, 4)$ and $S(6, 8)$, we can use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates of $Q$ and $S$: $d = \sqrt{(6 - 2)^2 + (8 - 4)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{meters}$

The distance between $Q$ and $S$ is $4\sqrt{2} \approx 5.66$ meters.

Part (c) - If point $G$ divides the line $QR$ in the ratio $1:2$, find the coordinate of $G$

To find the coordinates of a point that divides a line in a given ratio, we use the section formula. The section formula for dividing a line between points $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ is: $(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)$ Here, point $G$ divides line $QR$ in the ratio 1:2. The coordinates of $Q$ are $(2, 4)$, and the coordinates of $R$ are $(6, 4)$. Substituting into the section formula: $x_G = \frac{1 \times 6 + 2 \times 2}{1 + 2} = \frac{6 + 4}{3} = \frac{10}{3} \approx 3.33$ $y_G = \frac{1 \times 4 + 2 \times 4}{1 + 2} = \frac{4 + 8}{3} = \frac{12}{3} = 4$

Thus, the coordinates of point $G$ are $\left( \frac{10}{3}, 4 \right) \approx (3.33, 4)$.

Summary:

• (a) The width of rectangle $PQRS$ is 4 meters.
• (b) The distance between $Q$ and $S$ is $4\sqrt{2}$ meters or approximately 5.66 meters.
• (c) The coordinates of point $G$ are approximately $(3.33, 4)$.

Let me know if you'd like further clarification or additional details.

Related Questions:

1. How do you find the length of diagonal $PR$ using the distance formula?
2. What is the area of rectangle $PQRS$?
3. How would you calculate the perimeter of rectangle $PQRS$?
4. If point $G$ divides $PR$ instead of $QR$, what would the coordinates of $G$ be?
5. How does the section formula change if the ratio is reversed, i.e., $2:1$?

Tip: Remember that the section formula can be applied in both 2D and 3D geometry to find the coordinates of a dividing point.