https://file.aiquickdraw.com/mathbot/file/1726138478488lvvi2ola.jpg

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 \( \left( \frac{10}{3}, 4 \right) \approx (3.33, 4) \).

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### 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.

Sapling of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a rectangular grassy lawn in the plot as shown in the figure. The students are sowing seeds of flowering plants on the remaining area of the plot. (a) Find the width of rectangle PQRS. (b) Find the distance between the vertices of diagonal QS. (c) If point G divides the line QR in the ratio 1:2, then find the coordinates of G.

Solve the geometry problem involving rectangle PQRS. Learn how to find the width, the distance between diagonal points QS using the distance formula, and the coordinates of point G dividing QR using the section formula. Suitable for Grade 10 students.

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