Three stations A, B and C are at same level on the ground. Following observations were made from two stations B and C which are 70 m apart: (i) horizontal angle at station B between A and C = 60° and (ii) horizontal angle at station C between A and B = 30°. What is the distance between stations A and C?
- A35.0 m
- B49.5 m
- C60.6 mCorrect
- D70.0 m
Solution
Concept
Application of the Sine Rule in a plane triangle to find unknown sides when two angles and one side are known.
Formula
AC / sin(B) = BC / sin(A)
Step-by-step solution
1. Identify the triangle ABC with stations B and C 70 m apart (so side BC = 70 m). 2. Given horizontal angle at B (angle ABC) = 60° and horizontal angle at C (angle ACB) = 30°. 3. Calculate the third angle at A: angle BAC = 180° - 60° - 30° = 90°. 4. Apply the Sine Rule: AC / sin(ABC) = BC / sin(BAC). 5. Substitute the values: AC / sin(60°) = 70 / sin(90°). 6. AC = 70 × sin(60°) / 1 = 70 × 0.8660 = 60.62 m (approx 60.6 m).
Explanation
In triangle ABC, the sum of angles is 180°. Therefore, the angle at station A is 180° - 60° - 30° = 90°. Using the sine rule, AC / sin(B) = BC / sin(A), which gives AC = 70 × sin(60°) / sin(90°) = 70 × 0.866 / 1 = 60.62 m.
Why the other options are wrong
A: 35.0 m results from incorrect halving of the base. B: 49.5 m is an arithmetic error. D: 70.0 m confuses the base length with the side AC.
Shortcut
Recognize that the triangle is a right-angled triangle since the angles are 60°, 30°, and 90°. The side opposite to 90° is the hypotenuse (BC = 70 m). Then AC = BC × sin(60°) = 70 × 0.866 = 60.6 m.
Common mistake
Using the wrong angle in the sine rule numerator or incorrectly identifying which side is the 70 m base.
Correct answer: C) 60.6 m