Point S is 4 km west of point R, and point T is 4 km southwest of S. Find the bearing and distance from R to T.
(If possible, show diagrams too, please! Thanks in advance.)
T is 4/√2 km west and 4/√2 km south of S.
So, T is 4+4/√2 = 6.83 km west and 4/√2 = 2.83 km south of R.
Or, if you think of R as (0,0), T is (-6.83,-2.83)
arctan 2.83/6.83 = 22.49°
So, the bearing from R to T is W22.5°S
To find the bearing and distance from point R to point T, we can use trigonometry and the concept of vectors. Let's start by visualizing the scenario:
```
T
|\
| \
| \
| S
| |
| |
R______|___|
```
Given that point S is 4 km west of point R, we can draw a line segment of length 4 km from R to S in the west direction.
```
T
|\
| \
| \
| S
...----|---|----...
...|---|---...
R
```
Next, we are told that point T is 4 km southwest of point S. This means we need to draw a line segment from S to T, with a length of 4 km, in the southwest direction.
```
|\
| \
| \
\ T
\ /
\/
S
...----|---|----...
...|---|---...
|
|
R
```
Now that we have our diagram, we can find the bearing and distance from R to T.
First, let's calculate the distance. We can use the Pythagorean theorem since we have a right triangle formed by points R, S, and T. The horizontal length RS and vertical length ST form the legs of the triangle, and the hypotenuse is the distance from R to T.
Using the Pythagorean theorem:
RS² + ST² = RT²
Since RS is 4 km and ST is 4 km:
4² + 4² = RT²
16 + 16 = RT²
32 = RT²
Taking the square root of both sides: RT ≈ √32 ≈ 5.66 km
So, the distance from R to T is approximately 5.66 km.
Next, let's find the bearing from R to T. The bearing is the direction in which we would need to travel from R to reach T. We can use trigonometry to find the acute angle θ between RS (west) and RT (hypotenuse).
Using the tangent function:
tan(θ) = opposite/adjacent = ST/RS
tan(θ) = 4 km/4 km = 1
Taking the arctan of both sides: θ ≈ arctan(1)
θ ≈ 45°
Therefore, the bearing from R to T is approximately 45 degrees.
In summary, the bearing from point R to point T is approximately 45 degrees, and the distance is approximately 5.66 km.