A cyclist start from a point x and rides 3km due east to a point y. At y,he changes direction and rides 5km north west to a point z. Find the bearing of z from x to the nearest degree
To find the bearing of point z from point x, we need to determine the angle of the line connecting these two points with respect to the north direction.
First, we draw a diagram of the situation described:
N
|
|
|
|
Z ----X---- Y
E
We know that the cyclist rides 3km due east from point x to point y. This creates a right-angled triangle with its hypotenuse being the line XY. The angle between the line XY and the due east direction is 90°.
Next, the cyclist changes direction and rides 5km northwest from point y to point z. This creates a right-angled triangle with its hypotenuse being the line YZ. The northwest direction corresponds to the bearing of 315°.
Now, we can find the bearing of point z from point x by adding the two angles: 90° + 315° = 405°.
However, a bearing must be expressed within the range of 0° to 360°, so to find the bearing to the nearest degree, we need to subtract 360° from the calculated bearing: 405° - 360° = 45°.
Therefore, the bearing of point z from point x is 45° to the nearest degree.
To find the bearing of point z from point x, we need to determine the angle between the line from x to z and the north direction.
Step 1: Draw a diagram to visualize the situation. Place point x, point y (3 km east of x), and point z (5 km northwest of y).
z
/
/
5km /
/
/
y-----/
/
/ 3km
/
x
Step 2: Calculate the distance and direction from y to z. We can use the Pythagorean theorem to find the distance.
Distance = √(5km)^2 + (5km)^2
= √(25km^2 + 25km^2)
= √(50km^2)
= 5√2 km (approximately 7.07 km)
Step 3: Determine the direction or bearing of z from y. Since z is northwest of y, the bearing can be calculated as:
Bearing = 135°
Step 4: Determine the overall bearing of z from x. Since z is 3 km east of x and 5√2 km northwest of y, we need to consider the relative directions:
- The bearing from x to y is 90° (east).
- The bearing from y to z is 135° (northwest).
- Add these two angles: 90° + 135° = 225°.
Therefore, the bearing of z from x is approximately 225° to the nearest degree.