Three towns P,Q and R are such that the distance between P and Q is 50 km and the distance between P and R is 90km. if the bearing of Q from P is 075° and the bearing of R from P is 345°, find the distance between Q and R, and bearing of R from Q

If you sketch it , you know two sides, and the included angle. Law of cosines to get the third side QR, and then the law of sines to get the interior angles, and for the final bearing, you should be able to add the angles to figure it.

The bearing of two points Q and R from a points P are 030 degree and 130 degree respectively. If PQ = 12cm and PR=5cm, find the QR

I don't understand

To find the distance between Q and R and the bearing of R from Q, we can use trigonometry and geometry concepts. Let's break down the steps:

Step 1: Understand the problem.
We have three towns: P, Q, and R. The distance between P and Q is 50 km, and the distance between P and R is 90 km. We also know the bearings of Q from P and R from P.

Step 2: Draw a diagram.
We can sketch a diagram to visualize the positions of the towns and the given bearings. Let's draw three points, P, Q, and R, and label the distances and bearings.

Q
/
/
/
/
P -------------- R

Step 3: Analyze the given bearings.
The bearing of Q from P is 075°, which means that if we draw a line from P to Q, it forms a 75° angle with the north direction. Similarly, the bearing of R from P is 345°, which implies a 15° angle with the north direction.

Step 4: Calculate the angle between Q and R.
To find the angle between Q and R, we can subtract the bearing of Q from the bearing of R:
Angle (Q-R) = Bearing of R - Bearing of Q = 345° - 75° = 270°

Step 5: Use the Law of Cosines to find the distance between Q and R.
We can use the Law of Cosines to find the side length QR. In a triangle QPR, with side lengths PQ = 50 km, PR = 90 km, and angle PQR = 270°, the Law of Cosines states:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(PQR)

First, convert the angle PQR from degrees to radians:
PQR = (270° * π) / 180° = 3π/2 radians

Then, substitute the known values into the formula:
QR^2 = (50 km)^2 + (90 km)^2 - 2 * (50 km) * (90 km) * cos(3π/2)

Evaluate the cosine of 3π/2:
cos(3π/2) = 0

Simplify the equation:
QR^2 = 2500 km^2 + 8100 km^2 - 0
QR^2 = 10600 km^2

Take the square root of both sides to find QR:
QR = √10600 km
QR ≈ 103 km

Therefore, the distance between Q and R is approximately 103 km.

Step 6: Calculate the bearing of R from Q.
To find the bearing of R from Q, we need to determine the angle between the line QR and the north direction.

Since we already know the angle PQR is 270°, the angle between QR and the north direction is 360° - 270° = 90°.

Therefore, the bearing of R from Q is 90°.

Final answer:
The distance between Q and R is approximately 103 km, and the bearing of R from Q is 90°.