--- title: "Closed traverse ABCDA, B at 26500 N 22400 E" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:closed-traverse-abcda-missing-2066m" source: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-traverse-abcda-missing-2066m site: BCT Engineering --- # Closed traverse ABCDA, B at 26500 N 22400 E Surveying II · Chapter 1: Traversing - Asked in: 2066 Magh (old course), 10 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/closed-traverse-abcda-missing-2066m - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question In a four sided closed traverse ABCDA, the following informations are given: | Side | Length (m) | Deflection angle | Bearing | Northing | Easting | Remarks | |---|---|---|---|---|---|---| | AB | 160 | ? | S 40°00' W | ? | ? | | | BC | 340 | 116°00' (L) | ? | 26500 | 22400 | Coordinates of B | | CD | 210 | 60°00' (L) | ? | ? | ? | | | DA | ? | ? | ? | ? | ? | | Find the missing data. ## Short answer From left deflection angles, bearings are BC 104°00' and CD 44°00'. Closure gives DA length 376.788 m and bearing 278°12'; missing left deflections are 58°12' at A and 125°48' at D, summing with given angles to 360°. ## Answer The deflection angles are to the left, so the traverse runs anticlockwise and **bearing of a line = bearing of previous line − left deflection**. For a closed traverse $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$, which gives the length and bearing of the last line. #### Step 1: Bearings of the known lines - AB: S40°W = 220°00' - BC: 220°00' − 116°00' = 104°00' - CD: 104°00' − 60°00' = 44°00' #### Step 2: Latitudes and departures of the known lines $$ \text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta $$ | Line | L (m) | Bearing | Latitude | Departure | |---|---|---|---|---| | AB | 160.00 | 220°00' | -122.567 | -102.846 | | BC | 340.00 | 104°00' | -82.253 | +329.901 | | CD | 210.00 | 44°00' | +151.061 | +145.878 | | **Sum** | | | -53.759 | +372.933 | #### Step 3: Length and bearing of DA For closure, the missing line must cancel the sums: $$ \begin{aligned} \text{Lat}_{DA}&=-(-53.759)=+53.759\ \text{m}\\ \text{Dep}_{DA}&=-(+372.933)=-372.933\ \text{m}\\ L_{DA}&=\sqrt{(+53.759)^2+(-372.933)^2}=376.788\ \text{m}\\ \tan\theta&=\frac{-372.933}{+53.759}\ \Rightarrow\ \theta_{DA}=278^\circ 12' \end{aligned} $$ (The signs of the latitude and departure put the line in the north-west quadrant.) #### Step 4: Missing deflection angles Left deflection at a station = bearing of previous line − bearing of next line (add 360° if the result is negative). - At A (DA to AB): 278°12' − 220°00' = 58°12' - At D (CD to DA): 44°00' − 278°12' + 360° = 125°48' Check: sum of left deflections = 116°00' + 60°00' + 58°12' + 125°48' = 360°00' (anticlockwise traverse). #### Step 5: Coordinates (B = 26500 N, 22400 E) Independent coordinates = previous coordinates + latitude/departure of the line. | Station | Northing (m) | Easting (m) | |---|---|---| | A | 26622.567 | 22502.846 | | B | 26500.000 | 22400.000 | | C | 26417.747 | 22729.901 | | D | 26568.808 | 22875.779 | Check: A plus line AB gives (26500.000 N, 22400.000 E), which is the given point, so the traverse closes. #### Summary of the table | Line | Length (m) | Deflection at start (left) | Bearing (WCB) | Latitude | Departure | |---|---|---|---|---|---| | AB | 160.000 | 58°12' (computed) | 220°00' | -122.567 | -102.846 | | BC | 340.000 | 116°00' (given) | 104°00' | -82.253 | +329.901 | | CD | 210.000 | 60°00' (given) | 44°00' | +151.061 | +145.878 | | DA | 376.788 (computed) | 125°48' (computed) | 278°12' (computed) | +53.759 | -372.933 | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For this anticlockwise traverse with left deflections, the bearing of a line equals: A. Previous bearing + left deflection B. Previous bearing - left deflection C. Previous bearing + 180° - deflection D. Previous bearing - 180° Answer: B. A left deflection turns the line anticlockwise, reducing the whole circle bearing: BC = 220° - 116° = 104°. (Key point: Bearing from left deflection) 2. What is the whole circle bearing of AB given as S 40° W? A. 140° B. 40° C. 320° D. 220° Answer: D. S 40° W lies in the third quadrant: 180° + 40° = 220°. (Key point: Convert bearing) 3. How are the length and bearing of the last line DA found? A. It has the same length as AB B. Its latitude and departure are the negatives of the sums of the other three lines C. By adding the three bearings D. By measuring in the field Answer: B. For a closed traverse sum Lat = 0 and sum Dep = 0, so DA cancels the others: Lat = +53.759, Dep = -372.933, giving L = 376.788 m. (Key point: Closure condition) 4. DA has latitude +53.759 and departure -372.933. Its whole circle bearing is: A. 81°48' B. 98°12' C. 261°48' D. 278°12' Answer: D. Positive latitude and negative departure put the line in the NW quadrant, so WCB = 360° - 81°48' = 278°12'. (Key point: Quadrant of DA) ## Related in Surveying II - [Closed traverse missing EA bearing, BC length](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-traverse-abcdea-missing-ea-bc) — Surveying II, chapter 1 - [Closed traverse ABCDA missing figures (C=2500)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-traverse-abcda-missing-2072m) — Surveying II, chapter 1 - [Closed traverse ABCDA missing data, C(1200, 2000)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-traverse-abcda-c-1200-2000) — Surveying II, chapter 1 - [Closed traverse ABCDA, missing data](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-traverse-abcda-missing-2069b) — Surveying II, chapter 1 - [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 - [Gale's table for traverse ABCDA](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Agale-table-abcda-2072a) — Surveying II, chapter 1 ## Related in other subjects - [Design pipes AB, BC; 160 lpcd, peak factor 3](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Apd-c71) — Water Supply Engineering, chapter 7 - [Mean angles, traverse ABCDA (2076)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-horizontal-angles-abcda-hcr-90-00-20-c-b) — Surveying I, chapter 7, asked 2× - [Mean angles, traverse ABCDA (2068 Baisakh)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-horizontal-angles-abcda-hcr-209-25-40) — Surveying I, chapter 7, asked 2× - [Theodolite traverse ABCDA with missing VCR (68°30′10″)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-traverse-abcda-hcr-vcr-68-30-10) — Surveying I, chapter 7