---
title: "Intersection, coordinates of B from S and A"
subject: "Surveying II"
chapter: 5
key: "semester_4/surveying_ii:intersection-point-b-from-s-and"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/intersection-point-b-from-s-and
site: BCT Engineering
---

# Intersection, coordinates of B from S and A

Surveying II · Chapter 5: Orientation

- Asked in: 2071 Magh, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/intersection-point-b-from-s-and
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-5

## Question

The co-ordinates of stations S and A are (1309.12 m E, 1170.50 m N) and (1525.43 m E and 956.87 m N) respectively. Calculate the co-ordinates of point B which has been located by intersection from stations S and A observing the following angles. ∠BSA = 85°38'49" and ∠SAB = 55°50'33".

## Short answer

Base SA=304.019 m, bearing SA=134°38'34". In triangle SAB, ∠SBA=38°30'38", SB=404.034 m, AB=486.852 m. Bearing SB=48°59'45", so B = (1614.029 mE, 1435.592 mN).

## Answer

Coordinates are (Easting, Northing). B is taken on the left-hand (north-east) side of the line S to A; if it were on the other side, its position would be the mirror image of this one in the line SA.

```text
           B
          / \
         /   \
        /85°  \ 55°
       S ------ A
```

#### Step 1: Base SA

$$
\Delta E = 216.31,\quad \Delta N = -213.63
$$

$$
SA = \sqrt{216.31^2 + (-213.63)^2} = 304.019\ \text{m}, \qquad \text{bearing of SA} = 134°38'34"
$$

#### Step 2: Triangle SAB

$\angle SBA = 180^\circ - 85^\circ38'49'' - 55^\circ50'33'' = 38°30'38"$

$$
SB = \frac{SA\sin 55^\circ50'33''}{\sin 38°30'38"} = 404.034\ \text{m}, \qquad AB = \frac{SA\sin 85^\circ38'49''}{\sin 38°30'38"} = 486.852\ \text{m}
$$

#### Step 3: Coordinates of B

Bearing of SB = bearing of SA − 85°38'49" $= 48°59'45"$

$$
E_B = 1309.12 + 404.034\sin 48°59'45" = 1614.029\ \text{m}
$$

$$
N_B = 1170.50 + 404.034\cos 48°59'45" = 1435.592\ \text{m}
$$

**Check from A:** bearing of AB = bearing of AS + 55°50'33" $= 10°29'07"$, giving $E = 1614.029$ and $N = 1435.592$ (the same).

**Answer: coordinates of B = (1614.029 E, 1435.592 N).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the angle SBA at the new point?
   A. 85°38'49" + 55°50'33"
   B. 180° - 85°38'49" - 55°50'33" = 38°30'38"
   C. 360° - 85°38'49"
   D. 55°50'33" - 85°38'49"
   Answer: B. The triangle angles sum to 180°, so the angle at B is the remainder, 38°30'38". (Key point: Angle at B)

2. Which expression gives SB?
   A. SA sin 85°38'49" / sin 38°30'38"
   B. SA sin 55°50'33" / sin 38°30'38", since SB is opposite the angle at A
   C. SA sin 38°30'38"
   D. SA / sin 55°50'33"
   Answer: B. SB is opposite the angle at A (55°50'33"), while SA is opposite the angle at B. (Key point: Sine rule for SB)

3. B is on the left-hand side of the line S to A. The bearing of SB is:
   A. Bearing of SA + 85°38'49"
   B. Bearing of SA - 85°38'49" = 48°59'45"
   C. Bearing of SA + 180°
   D. 85°38'49"
   Answer: B. The left-hand side is anticlockwise from SA, so the angle is subtracted from the bearing of SA (134°38'34"). (Key point: Bearing of SB)

4. How is the position of B verified?
   A. By computing B again from A using bearing AB = bearing AS + 55°50'33" and getting the same coordinates
   B. By repeating the same calculation from S
   C. By measuring SB with a tape
   D. By adding the sine values
   Answer: A. An independent computation from A gives E = 1614.029, N = 1435.592, agreeing with the result from S. (Key point: Check from the second station)

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