To solve the problem, we need to visualize the situation and apply some trigonometry. Let's break it down step-by-step. ### Step 1: Understanding Bearings Bearings are measured clockwise from the north direction. - The bearing of point Q from point P is **N67°W**. This means if you start facing north and turn 67° towards the west, you will be facing point Q. - The bearing of point R from point P is **N23°E**. This means if you start facing north and turn 23° towards the east, you will be facing point R. - The bearing of point R from point Q is **N68°E**. This means if you are at point Q, you would turn 68° towards the east to face point R. ### Step 2: Drawing the Diagram 1. **Draw a vertical line** to represent the north direction. 2. **From point P**, draw a line towards Q at an angle of 67° to the west of north. This line represents PQ. 3. **From point P**, draw another line towards R at an angle of 23° to the east of north. This line represents PR. 4. **From point Q**, draw a line towards R at an angle of 68° to the east of north. ### Step 3: Labeling the Diagram - Let’s denote the angle at point P between PQ and PR as angle \( \theta \). - The angle between the north line and line PQ is 67°, and the angle between the north line and line PR is 23°. - Therefore, the angle \( \theta \) at point P can be calculated as: \[ \theta = 67° + 23° = 90° \] This means that the lines PQ and PR are perpendicular to each other. ### Step 4: Using the Right Triangle Since we have established that PQ and PR are perpendicular, we can use the Pythagorean theorem to find the length of PR. 1. We know: - \( PQ = 150 \, m \) - \( QR \) can be found using the bearing from Q to R, which is N68°E. 2. To find \( QR \), we can use the sine rule or cosine rule, but since we have a right triangle, we can directly find the lengths using trigonometric ratios. ### Step 5: Finding PR Using the right triangle formed by points P, Q, and R: - We can use the sine of the angle at Q (which is 68°) to find QR: \[ QR = PQ \cdot \tan(68°) \] However, we need to find PR directly. Since we have a right triangle, we can use the cosine rule: \[ PR^2 = PQ^2 + QR^2 \] But we need to find QR first. ### Step 6: Calculate QR Using the sine rule: \[ QR = PQ \cdot \tan(68°) = 150 \cdot \tan(68°) \] Calculating \( \tan(68°) \): \[ \tan(68°) \approx 2.475 \] Thus, \[ QR \approx 150 \cdot 2.475 \approx 371.25 \, m \] ### Step 7: Calculate PR Now, we can find PR using the Pythagorean theorem: \[ PR^2 = PQ^2 + QR^2 \] Substituting the values: \[ PR^2 = 150^2 + 371.25^2 \] Calculating: \[ PR^2 = 22500 + 137134.0625 \approx 159634.0625 \] Taking the square root: \[ PR \approx \sqrt{159634.0625} \approx 399.54 \, m \] ### Conclusion After calculating, we find that the length of PR is approximately 399.54 m, which does not match any of the options provided. ### Revision Summary - Bearings are measured clockwise from north. - Use trigonometric ratios to find lengths in right triangles. - The Pythagorean theorem is useful for calculating distances in right triangles. - Always double-check calculations for accuracy. ### Final Answer The correct option is not listed among A, B, C, or D based on the calculations provided. Please verify the problem statement or the options given.