A village of P is 10km from a lorry station Q others bearing 065degrees .Another village R is 8km from Q on the bearing 155 . Calculate
The distance of R from p,to the nearest km
The bearing of R from P to the nearest degree
M is a village on PR such that QMis perpendicular to PR. fine the distance of M from P to the nearest km
No response
notice that angle PQR is 90°
so PR^2 = 10^2 + 8^2
find the bearing in the usual way.
Then, using similar triangles, MP/10 = 10/PR
I don't understand
Where is the answer
To solve this problem, we can use trigonometry and the concept of bearings. Let's break it down step by step:
1. Find the distance of R from P:
We can use the cosine rule to find the distance between two points given the lengths of two sides and the included angle. In this case, we know the lengths of sides PQ, PR, and the included angle QPR.
Using the cosine rule:
PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)cos(angle QPR)
PQ = 10 km
QR = 8 km
angle QPR = 90 degrees (since QM is perpendicular to PR)
PR^2 = 10^2 + 8^2 - 2(10)(8)cos(90)
PR^2 = 100 + 64 - 160
PR^2 = 4
Taking the square root of both sides, PR = 2 km (nearest km)
2. Find the bearing of R from P:
The bearing is the angle measured clockwise from the north direction. To find the bearing of R from P, we need to find the angle PRQ, which is the difference between the bearing of Q from P and the bearing of R from Q.
The bearing of Q from P is 65 degrees. Since the bearing is measured clockwise, the bearing of R from P is 65 + 155 = 220 degrees (nearest degree).
3. Find the distance of M from P:
Since QM is perpendicular to PR, we have a right triangle QMP. Given the lengths of QM (which we can calculate as PQ - QR) and QP (which is 10 km), we can use Pythagoras' theorem to find the length of MP.
QM = PQ - QR
QM = 10 km - 8 km = 2 km
Applying Pythagoras' theorem:
MP^2 = QM^2 + QP^2
MP^2 = 2^2 + 10^2
MP^2 = 4 + 100
MP^2 = 104
Taking the square root of both sides, MP ≈ 10.2 km (nearest km)
Therefore, the answers are:
- The distance of R from P is 2 km (nearest km).
- The bearing of R from P is 220 degrees (nearest degree).
- The distance of M from P is approximately 10.2 km (nearest km).