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. What is the distance and bearing of the Red house from Green house?
I will interpret "bearing of 230°" to mean -140° in standard trig notation
Using vectors:
vector RG = (4,0) + (0,-3) + (4cos-140°, 4sin-40°)
= (4,-3) + (-3.0412, -2.57115)
= (.9358, -5.5712)
|RG| = √(.9358^2 + (-5.5712)^2) = appr 5.65 km
direction of RG :
tanØ = -5.5712/.9358
Ø = -80.46°
So in my diagram , that would make the angle between GR and a vertical equal to 9.54°
and using your notation , a bearing of 350.46°
No diagram?
To find the distance and bearing of the Red house from the Green house, we need to break down the given information and calculate the respective distances and bearings step by step.
1. Start with the given information:
- Blue house is 4 km due East of Red house.
- Yellow house is 3 km due South of Blue house.
- Green house is 4 km on a bearing of 230° from the Yellow house.
2. Draw a diagram to visualize the situation:
- Place the Red house as a starting point (you can label it R) arbitrarily.
- Draw a line eastward from the Red house and mark a point labeled Blue (4 km away).
- Draw a line southward from the Blue house and label a point labeled Yellow (3 km away).
- Finally, draw a line extending at an angle of 230° from the Yellow house and label the point of intersection as Green (4 km away).
R (Red) G (Green)
| ^
| |
| Yellow |
| | | 4 km
| v |
--> B (Blue) -------------------
3. Calculate the position of Green house relative to Red house:
- Since Blue is 4 km due East of Red, the coordinates of Blue house would be (4 km, 0 km).
- Since Yellow is 3 km due South of Blue, the coordinates of Yellow house would be (4 km, -3 km).
- To find the coordinates of Green house from Yellow house, we need to use the angle and distance information.
- The bearing angle of 230° means we move 230° counterclockwise from the positive x-axis.
- Convert the bearing angle to radians: 230° * (π / 180°) ≈ 4.01 radians.
- Use trigonometry to find the coordinates of Green house:
- x-coordinate: 4 km + 4 km * cos(4.01) ≈ -0.93 km (approximately -1 km, but it is eastward)
- y-coordinate: -3 km + 4 km * sin(4.01) ≈ -6.9 km (approximately -7 km, but it is southward)
4. Find the distance between Red and Green houses:
- We can use the distance formula (Pythagorean theorem) to find the distance between two points in a plane.
- Distance = √[(x2 - x1)² + (y2 - y1)²]
- Substituting the coordinates of Red and Green houses:
- Distance = √[(-1 km - 0 km)² + (-7 km - 0 km)²] ≈ √(1 km² + 7 km²) ≈ √(1 km² + 49 km²) ≈ √50 km ≈ 7.07 km (approximately)
5. Find the bearing angle of Green house from Red house:
- We can use trigonometry to find the angle between two points in a plane.
- Bearing = arctan((y2 - y1) / (x2 - x1))
- Substituting the coordinates of Red and Green houses:
- Bearing = arctan((-7 km - 0 km) / (-1 km - 0 km)) ≈ arctan(-7 km / -1 km) ≈ arctan(7) ≈ 81.87° (approximately)
Therefore, the distance from the Red house to the Green house is approximately 7.07 km, and the bearing (angle) from the Red house to the Green house is approximately 81.87°.