https://file.aiquickdraw.com/mathbot/file/1726994988382wdj8lsa2.jpg

Let's work through each of the questions based on the provided figure and its details.

### 1. Taking A as the origin, find the coordinates of \( P \):
The rectangular plot is defined with points \( A \), \( B \), \( C \), and \( D \). From the saplings marked on the boundary, the grid can be used to identify the position of point \( P \).

From the figure, the distance between consecutive saplings appears to be 1 unit. By counting the positions, we can estimate the coordinates of \( P \). \( P \) seems to be at (4, 6) based on the alignment.

So, the answer is **(4, 6)**.

### 2. What will be the coordinates of \( R \), if \( C \) is the origin?
In this case, when \( C \) is the origin, we need to measure distances relative to point \( C \). \( R \) seems to be directly beneath \( C \) and a little to the right. Counting from \( C \), \( R \) should be at **(8, 6)**.

Thus, the answer is **(8, 6)**.

### 3. What will be the coordinates of \( Q \), if \( C \) is the origin?
To find \( Q \), we count from \( C \) again. \( Q \) is positioned closer to \( C \) in the opposite direction of \( R \). By counting the grid points, the coordinates of \( Q \) are approximately **(13, 6)** when \( C \) is the origin.

Thus, the answer is **(13, 6)**.

### 4. Calculate the area of the triangle if \( A \) is the origin.
The triangle formed is between \( A \), \( P \), and \( Q \). To calculate the area of the triangle using coordinates, we can apply the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Using the coordinates \( A(0, 0) \), \( P(4, 6) \), and \( Q(8, 6) \):

\[
\text{Area} = \frac{1}{2} \left| 0(6 - 6) + 4(6 - 0) + 8(0 - 6) \right|
= \frac{1}{2} \left| 0 + 24 - 48 \right| = \frac{1}{2} \times 24 = 12
\]

Thus, the area is **6** square units (option b).

### 5. Calculate the area of the triangle if \( C \) is the origin:
In this case, the triangle's vertices shift, but the base and height remain the same due to the symmetry of the rectangular plot. The area will still be the same as the previous calculation.

Thus, the answer is also **6**.

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Would you like to discuss further details on any part of this solution?

Here are 5 related questions to consider:
1. What is the formula for calculating the area of a triangle using coordinates?
2. How would the coordinates change if the origin were at point \( D \)?
3. Can you calculate the distance between \( P \) and \( Q \)?
4. What is the perimeter of the triangle formed by \( A \), \( P \), and \( Q \)?
5. How would the area change if the dimensions of the rectangular plot were doubled?

**Tip:** When working with coordinate geometry, symmetry and distance between points can often simplify calculations.

The class X students school in Krishnagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot. The students are to sow seeds of flowering plants in the remaining area. Based on this, solve the following coordinate geometry problems:

Solve problems related to coordinate geometry, including finding coordinates of points P, Q, and R, and calculating the area of triangles in a rectangular plot. Suitable for Grade 10 students learning about area calculations on the Cartesian plane.

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