The bearing of two lines AB and AC measured by using a surveyor’s compass are S 26° 40′ E and N 18° 30′ W, respectively. The value of ∠BAC measured in a clockwise direction is

🧭 Understanding the Core Concept

This problem requires calculating the included angle between two lines (AC and AB) given their bearings from a common point (A). The key is to visualize the lines on a compass rose and calculate the total angle traversed when moving clockwise from the starting line (AC) to the ending line (AB).

🔬 Step-by-Step Calculation

To find the clockwise angle ∠BAC, we can break the calculation into two parts based on the cardinal directions.

• Bearing of AC: N 18° 30' W (Line is in the North-West quadrant).
• Bearing of AB: S 26° 40' E (Line is in the South-East quadrant).

Step 1: Calculate the clockwise angle from line AC to the South line.

The angle from the North line to the South line is 180°. Since AC is 18° 30' West of North, the angle from AC to the South line in a clockwise direction is:

Angle = 180° - 18° 30' = 161° 30'

Step 2: Calculate the clockwise angle from the South line to line AB.

The bearing S 26° 40' E means the line AB is 26° 40' away from the South line, moving towards the East (which is a clockwise direction from South). Therefore, this angle is simply:

Angle = 26° 40'

Step 3: Sum the angles to find the total clockwise angle ∠BAC.

Add the results from Step 1 and Step 2:

∠BAC = 161° 30' + 26° 40' = 187° 70'

Step 4: Convert minutes to degrees.

Since 60 minutes (') equals 1 degree (°), we convert 70' to 1° 10'.

∠BAC = 187° + 1° 10' = 188° 10'

🗺️ Visualizing the Angle

The diagram below illustrates the positions of lines AC and AB and the clockwise angle between them.