From a point R, 300m north of P, a man walks eastwards to a place; Q which is 600m from P. Find the bearing of P from Q correct to the nearest degree

1. Analyze the question

We are given two points, P and R. Point R is 300m North of P. A man walks East from R to a point Q, such that the distance from P to Q is 600m. We need to find the bearing of P from Q.

2. Explain the concept

• Geometry: We have a right-angled triangle $\triangle PRQ$, where $\angle PRQ = 90^\circ$ (because R is North of P and Q is East of R).
• Trigonometry: In a right-angled triangle, we can use SOH-CAH-TOA.
• Bearing: The bearing of P from Q is the angle measured clockwise from the North line at Q to the line QP.

3. Show the solution

Step 1: Visualize the triangle

• $PR = 300\text{m}$ (Vertical side)
• $PQ = 600\text{m}$ (Hypotenuse)
• $\angle RPQ$ is the angle at P inside the triangle.

Using trigonometry in $\triangle PRQ$: $\cos(\angle RPQ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{PR}{PQ}$ $\cos(\angle RPQ) = \frac{300}{600} = 0.5$ $\angle RPQ = \cos^{-1}(0.5) = 60^\circ$

Step 2: Determine the bearing

• At point Q, draw a North line. Since Q is East of R, the line RQ is horizontal (West-East).
• The angle of the line QP relative to the South line at Q is equal to the alternate angle $\angle RPQ = 60^\circ$.
• The bearing is measured clockwise from the North line at Q.
• The angle from North to South is $180^\circ$.
• The bearing of P from Q is $180^\circ + (\text{angle between South and QP})$.
• Since the angle between the vertical (South) and QP is $60^\circ$, the bearing is $180^\circ + 60^\circ = 240^\circ$.

4. Identify the answer

The calculated bearing is $240^\circ$. Looking at the options provided: A. $026^\circ$ B. $045^\circ$ C. $210^\circ$ D. $240^\circ$

The correct option is D.

5. Tip

When dealing with bearing questions, always draw a neat sketch. Remember that "North" is always $000^\circ/360^\circ$, "East" is $090^\circ$, "South" is $180^\circ$, and "West" is $270^\circ$. Using alternate angles (the "Z" rule) is the fastest way to transfer angles between your North lines!