To solve this problem, we need to find the bearing of point R from point Q, given that P is equidistant from Q and R. Let's proceed step-by-step:
1. Identify the given bearings:
- The bearing of Q from P is 80°.
- The bearing of R from P is 130°.
2. Understand the geometry of the problem:
- Since P is equidistant from Q and R, triangle PQR is isosceles.
- Bearings are measured clockwise from the north direction.
3. Determine the internal angle [tex]\( \angle PQR \)[/tex]:
- To find [tex]\( \angle PQR \)[/tex], we calculate the difference between the bearings of Q and R from P.
- The internal angle [tex]\( \angle PQR = \text{bearing of R from P} - \text{bearing of Q from P} \)[/tex].
- Hence, [tex]\( \angle PQR = 130° - 80° = 50° \)[/tex].
4. Internal angle at point Q ([tex]\( \angle RQP \)[/tex]):
- Since triangle PQR is isosceles and P is equidistant from Q and R, the internal angle RQP will also include the same value as [tex]\( \angle PQR \)[/tex]. Therefore, [tex]\( \angle RQP = 50° \)[/tex].
5. Determine the bearing of R from Q:
- To find the bearing of R from Q, we need to account for the direction R is in relation to Q.
- Since the bearing of Q from P was 80°, and the bearing of P from Q would be 80° + 180° = 260° (since reversing direction adds 180°).
- Therefore, to get the bearing of R from Q, we take 360° (full circle) and subtract the internal angle [tex]\( \angle RQP \)[/tex] because bearing is in clockwise direction.
- So, the bearing of R from Q = 360° - 50° = 310°.
Thus, the final answer is:
- The internal angle [tex]\( \angle PQR = 50° \)[/tex]
- The bearing of R from Q = 310°
So, the bearing of R from Q is [tex]\( \boxed{310°} \)[/tex].
