Houses are arranged in a school compound. Thus, Blue house is 4 km due East of Red, Yellow is 3 km due south of Blue and Green is 4 km on a bearing of 230° from the Yellow. Calculate the distance and bearing of the Red house from Green house.

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To calculate the distance and bearing of the Red house from the Green house, we'll need to break down the steps and use basic concepts of trigonometry and bearings.

Step 1: Draw a diagram
We start by drawing a diagram to visualize the given information.

- Place the Red house at a certain location.
- Draw a line for the Blue house that is 4 km due East from the Red house.
- Draw another line for the Yellow house that is 3 km due South from the Blue house.
- Finally, draw a line for the Green house that is 4 km on a bearing of 230° from the Yellow house.

Step 2: Calculate the position of each house
Based on the given directions, we can determine the approximate coordinates of each house. Let's assume the Red house is located at the origin (0,0).

- Red house: (0, 0)
- Blue house: (4, 0) (4 km due East from Red)
- Yellow house: (4, -3) (3 km due South from Blue)
- Green house: ? (4 km on a bearing of 230° from Yellow)

Step 3: Calculate the coordinates of the Green house
To find the coordinates of the Green house, we need to convert the bearing of 230° to a vector form. With trigonometry, we can break it down into its X and Y components.

- Bearing of 230° means the angle is measured clockwise from the North, so subtract it from 360° to get 130°.
- The X-component can be found using cosine: X = distance * cos(angle)
- The Y-component can be found using sine: Y = distance * sin(angle)

Substituting the values:
X = 4 km * cos(130°)
Y = 4 km * sin(130°)

Calculating the X and Y coordinates, we find:
X ≈ -2.41 km
Y ≈ -3.52 km

Therefore, the Green house is approximately located at (-2.41, -3.52).

Step 4: Calculate the distance and bearing between Red and Green houses
Now that we have the coordinates of both houses, we can calculate the distance and bearing between them using the distance formula and the bearing formula.

- Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
- Bearing formula: θ = atan2(y₂ - y₁, x₂ - x₁)

Substituting the values:
d = √((-2.41 - 0)² + (-3.52 - 0)²) km
d ≈ √(5.8081 + 12.3904) km
d ≈ √18.1985 km
d ≈ 4.27 km (rounded to two decimal places)

θ = atan2((-3.52 - 0), (-2.41 - 0))
θ ≈ atan2(-3.52, -2.41)

Calculating the bearing using a calculator or trigonometric functions, we find:
θ ≈ -125.16°

Therefore, the distance from the Red house to the Green house is approximately 4.27 km, and the bearing is approximately 125.16°.

Note: Bearings are typically measured clockwise from the North direction. In this case, since the bearing is negative, we can add 360° to get the bearing with a positive value:
θ ≈ 234.84°

Thus, the bearing of the Red house from the Green house is approximately 234.84°.