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Given the bearing of P from Q is 228^ and QPR = 72^, and R is due south of Q, find the bearing of Q from P and the bearing of P from R.

Mathematics

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Answers

2 parts
a

048∘048^\circ

b

060∘060^\circ

Step-by-step solution

5 steps

Here's how to solve the bearing problem:

Part a· 2 steps

Find the bearing of Q from P

  1. Identify the given bearing.

    The bearing of P from Q is 228∘228^\circ.

  2. Calculate the back bearing.

    To find the bearing of Q from P, we use the rule for back bearings: if the bearing of A from B is θ\theta, then the bearing of B from A is θ±180∘\theta \pm 180^\circ. Since 228∘>180∘228^\circ > 180^\circ, we subtract 180∘180^\circ.

    Bearing of Q from P=228∘−180∘=48∘\text{Bearing of Q from P} = 228^\circ - 180^\circ = 48^\circ

    The bearing of Q from P is 048∘\boxed{048^\circ}.

Part b

060∘060^\circ

Find the bearing of P from R

  1. Determine the angle ∠PQR\angle PQR.

    We are given that R is due south of Q. This means the line segment QR lies on the North-South line passing through Q.
    The bearing of P from Q is 228∘228^\circ. This angle is measured clockwise from the North line at Q to the line QP.
    The angle from the North line at Q to the South line at Q is 180∘180^\circ.
    Therefore, the angle between the South line (QR) and the line QP, which is ∠PQR\angle PQR, is:

    ∠PQR=228∘−180∘=48∘\angle PQR = 228^\circ - 180^\circ = 48^\circ
  2. Determine the angle ∠PRQ\angle PRQ in triangle PQR.

    We know the sum of angles in a triangle is 180∘180^\circ.
    Given ∠QPR=72∘\angle QPR = 72^\circ.
    From Step 1, ∠PQR=48∘\angle PQR = 48^\circ.
    So, ∠PRQ=180∘−∠QPR−∠PQR\angle PRQ = 180^\circ - \angle QPR - \angle PQR

    ∠PRQ=180∘−72∘−48∘\angle PRQ = 180^\circ - 72^\circ - 48^\circ ∠PRQ=180∘−120∘\angle PRQ = 180^\circ - 120^\circ ∠PRQ=60∘\angle PRQ = 60^\circ
  3. Determine the bearing of P from R.

    Since R is due south of Q, the line segment RQ points North from R. This means the North line at R is along RQ.
    The bearing of P from R is the angle measured clockwise from the North line at R (which is RQ) to the line segment RP.
    This angle is exactly ∠PRQ\angle PRQ.
    Therefore, the bearing of P from R is 60∘60^\circ. As a three-figure bearing, it is 060∘060^\circ.
    The bearing of P from R is 060∘\boxed{060^\circ}.

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