Sketch the following plot of land and find its area. From a cement marker, proceed 260m southwest to the river, then 240m south along the river to the bridge, then 280m N40°E to a sign on the edge of Sycamore Lane and finally along Sycamore Lane back to the cement marker.

Enter answer rounded to the thousandths.

To solve this problem, we need to interpret the path traveled on the plot of land and find its area based on the given directions. Let's outline and calculate it step-by-step.

### Step 1: Breakdown of the Directions
The path starts at a cement marker and proceeds through four segments:

1. **First Segment**: 260 meters southwest to the river.
   - Southwest means 45° south of west, or an angle of 225° from the north.
   
2. **Second Segment**: 240 meters south along the river to the bridge.

3. **Third Segment**: 280 meters N40°E (40° north of east) to a sign on the edge of Sycamore Lane.

4. **Fourth Segment**: Back to the cement marker, along Sycamore Lane.

### Step 2: Setting Up Coordinate System
We'll define the starting point (cement marker) as the origin \((0, 0)\). Let's use a coordinate system where:
- \( x \)-axis represents east/west direction.
- \( y \)-axis represents north/south direction.

**First Segment**: \( 260 \) meters southwest.
- Southwest is at a \( 225^\circ \) angle, so the coordinates are:
\[
x_1 = 260 \cos(225^\circ) = -183.85 \text{ meters}
\]
\[
y_1 = 260 \sin(225^\circ) = -183.85 \text{ meters}
\]
So, the endpoint of the first segment is at \((-183.85, -183.85)\).

**Second Segment**: \( 240 \) meters south.
- Moving directly south from the point \((-183.85, -183.85)\):
\[
y_2 = -183.85 - 240 = -423.85 \text{ meters}
\]
So, the endpoint is at \((-183.85, -423.85)\).

**Third Segment**: \( 280 \) meters N40°E.
- \( 40^\circ \) north of east:
\[
x_3 = -183.85 + 280 \cos(40^\circ) = -183.85 + 214.58 = 30.73 \text{ meters}
\]
\[
y_3 = -423.85 + 280 \sin(40^\circ) = -423.85 + 179.94 = -243.91 \text{ meters}
\]
So, the endpoint is at \((30.73, -243.91)\).

**Fourth Segment**: Back to the cement marker.
- We return to the origin \((0, 0)\).

### Step 3: Calculating the Area
To find the area of the plot, we can use the **shoelace formula** for the polygon with the four points:

1. \((0, 0)\)
2. \((-183.85, -183.85)\)
3. \((-183.85, -423.85)\)
4. \((30.73, -243.91)\)

**Shoelace Formula**:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1}) + x_n y_1 - \sum_{i=1}^{n-1} (y_i x_{i+1}) - y_n x_1 \right|
\]

Plugging in the points:

\[
\text{Area} = \frac{1}{2} \left| (0 \times -183.85) + (-183.85 \times -423.85) + (-183.85 \times -243.91) + (30.73 \times 0) - (0 \times -183.85) - (-183.85 \times -183.85) - (-423.85 \times 30.73) - (-243.91 \times 0) \right|
\]

I'll compute the exact value now.The area of the plot of land is approximately **50,995.882 square meters**.

Would you like more details or have any questions?

### Related Questions
1. How would the area change if the directions were slightly modified?
2. Can you show how to convert the directions into a graphical sketch?
3. What if we changed the reference angles to other directions (e.g., E30°S)?
4. How do you calculate distances between points using trigonometry?
5. How does the shoelace formula apply to non-quadrilateral shapes?

**Tip**: For irregular polygons, always verify the coordinates and use the shoelace formula carefully to ensure accuracy.

Sketch the following plot of land and find its area. From a cement marker, proceed 260m southwest to the river, then 240m south along the river to the bridge, then 280m N40°E to a sign on the edge of Sycamore Lane and finally along Sycamore Lane back to the cement marker.

Calculate the area of a plot of land by interpreting directions and using trigonometry and the shoelace formula. The plot involves moving 260m southwest, 240m south along the river, and 280m N40°E. Ideal for students in Grades 10-12.

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