---
title: "Length and bearing of CD from stations A, B"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tacheometer-length-bearing-cd-from-a"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-length-bearing-cd-from-a
site: BCT Engineering
---

# Length and bearing of CD from stations A, B

Surveying II · Chapter 2: Tacheometry

- Asked in: 2068 Bhadra, 9 marks
- Video: [How to calculate length and Bearing for theodolite Surveying](https://www.youtube.com/watch?v=r2sV9J1dQNE) — Mahesh More
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-length-bearing-cd-from-a
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

The following observations were taken from the traverse station A and B. The staff was held vertical. The tachometer is fitted with anallatic lens. Multiplicative constant = 100.

| Traverse station | H.I. (m) | Staff Station | Bearing | Vertical angle | Staff reading |
|---|---|---|---|---|---|
| A | 1.50 | C | 15°14' | +8°9' | 2.60, 1.85, 1.10 |
| B | 1.53 | D | 340°18' | +2°3' | 2.50, 1.91, 1.32 |

Independent coordinates of A is (800, 1800)
Independent coordinates of B is (950, 2500)
Compute the length and bearing of CD.

## Short answer

With anallatic lens, K=100, C=0, staff intercepts AC=1.500 m, BD=1.180 m. D=Ks cos²θ gives 146.985 m and 117.849 m. Coordinates: C (941.821, 1838.620), D (1060.951, 2460.274). Length CD = 632.965 m, bearing = 79°09′ (N 79°09′ E).

## Answer

Anallatic lens: $C=0$, $K=100$, staff vertical. The coordinates are taken as (Northing, Easting): A (800, 1800) and B (950, 2500). Only the horizontal distances $D$ are needed (the heights of instrument do not enter), so each sight gives the position of the staff point from its station.

$$
D=Ks\cos^2\theta
$$

#### Step 1: Distances AC and BD
**Point C** (bearing 15°14', angle of elevation $8^\circ 09'$):

$$
\begin{aligned}
s&=2.600-1.100=1.500\ \text{m}\\
D&=100\times1.500\times\cos^2 8^\circ 09'=146.985\ \text{m}\\
V&=\tfrac12\times100\times1.500\times\sin 2(8^\circ 09')=21.050\ \text{m}
\end{aligned}
$$

**Point D** (bearing 340°18', angle of elevation $2^\circ 03'$):

$$
\begin{aligned}
s&=2.500-1.320=1.180\ \text{m}\\
D&=100\times1.180\times\cos^2 2^\circ 03'=117.849\ \text{m}\\
V&=\tfrac12\times100\times1.180\times\sin 2(2^\circ 03')=4.218\ \text{m}
\end{aligned}
$$

#### Step 2: Coordinates of C and D
$$
\begin{aligned}
N_C&=800+146.985\cos15^\circ14'=941.821\\
E_C&=1800+146.985\sin15^\circ14'=1838.620\\
N_D&=950+117.849\cos340^\circ18'=1060.951\\
E_D&=2500+117.849\sin340^\circ18'=2460.274
\end{aligned}
$$

#### Step 3: Length and bearing of CD
$$
\begin{aligned}
\Delta N&=N_D-N_C=+119.130,\qquad \Delta E=E_D-E_C=+621.653\\
CD&=\sqrt{119.130^2+621.653^2}=632.965\ \text{m}\\
\theta_{CD}&=\tan^{-1}\frac{+621.653}{+119.130}=79^\circ 09'06''
\end{aligned}
$$

**Answer:** length of CD = **632.965 m**, bearing of CD = **79°09'** (N 79°09' E).

If the coordinates are instead read as (Easting, Northing), the same method gives CD = 672.956 m with a bearing of 6°06'44".

## Check yourself

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

1. Which quantity from each sight is needed to find coordinates of C and D?
   A. Horizontal distance D and bearing
   B. Only V
   C. Only HI
   D. Only the RL
   Answer: A. Plan coordinates need D and the bearing; heights are irrelevant to CD length and bearing. (Key point: Quantity needed)

2. How are N and E of C computed?
   A. N = N_A + D sin(bearing), E = E_A + D cos
   B. N = N_A + D cos(bearing), E = E_A + D sin(bearing)
   C. N = N_A + D, E = E_A + D
   D. N = N_A - D cos, E = E_A - D sin
   Answer: B. Latitude = D cos(bearing), departure = D sin(bearing) from the station. (Key point: Coordinates of staff point)

3. C is observed from A and D from B. What follows?
   A. Both are located from A
   B. The traverse must be closed first
   C. Each staff point is located from its own station coordinates
   D. The angle at A is used
   Answer: C. C uses A (800, 1800) and D uses B (950, 2500). (Key point: Different stations)

4. How are CD length and bearing found?
   A. √(ΔN + ΔE) and tan^-1(ΔN/ΔE)
   B. Difference of the two bearings
   C. D_C + D_D
   D. √(ΔN^2 + ΔE^2) and tan^-1(ΔE/ΔN) with ΔN = N_D - N_C, ΔE = E_D - E_C
   Answer: D. Standard inverse computation, giving 632.965 m and 79°09'. (Key point: Length and bearing of CD)

## Related in Surveying II

- [Anallatic tacheometer distance CD and gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-staff-cd-gradient-abcd) — Surveying II, chapter 2
- [Anallatic tacheometer AB gradient from P](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-ab-gradient-percent-angle) — Surveying II, chapter 2, asked 3×
- [Anallatic tacheometer distance XY, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-xy-gradient-p-2081c) — Surveying II, chapter 2, asked 3×
- [Anallatic tacheometer from O, RLs of A and B](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometer-o-distance-ab-rl-gradient) — Surveying II, chapter 2
- [Anallatic tacheometer distance XY, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-xy-gradient-p) — Surveying II, chapter 2, asked 3×
- [Traverse missing length CD, bearing EA](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-missing-cd-length-ea-bearing) — Surveying II, chapter 1

## Related in other subjects

- [Reciprocal levelling; RL of closed traverse stations](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-closed-traverse-bcdefgb) — Surveying I, chapter 5
- [Mean angles and VCR, station O (reading 179°59′40″)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-direction-o-abcd-first-reversed-179-59-40) — Surveying I, chapter 7, asked 3×
- [Double mass curve, station X and 15 stations, 2013-2022](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Admc-x-15stn-2013-2022) — Engineering Hydrology, chapter 2