--- title: "Traverse missing length CD, bearing EA" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:traverse-missing-cd-length-ea-bearing" source: https://www.bctengineer.com/q/semester_4/surveying_ii/traverse-missing-cd-length-ea-bearing site: BCT Engineering --- # Traverse missing length CD, bearing EA Surveying II · Chapter 1: Traversing - Asked in: 2078 Poush, 8 marks - Video: [omitted measurement of closed traverse || omitted measurement|| missing length & bearing in traverse](https://www.youtube.com/watch?v=iBV3daNWtaY) — Aim 2 Civil Class - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/traverse-missing-cd-length-ea-bearing - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question From the traverse data given below, find the missing length of line CD and bearing of line EA. | Line | AB | BC | CD | DE | EA | |---|---|---|---|---|---| | Length (m) | 178.60 | 228.40 | ? | 126.70 | 238.80 | | Bearing | S52°30'E | N48°45'E | N18°15'W | S78°30'W | ? | ## Short answer Compute latitudes/departures of known lines, solve closure equations. CD = 194.567 m; EA bearing = 212°30' (S32°30'W). ## Answer For a closed traverse $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$. Two quantities are unknown, the length of CD and the bearing of EA, so the two closure equations are enough to find them. #### Step 1: Latitudes and departures of the known lines $$ \text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta $$ | Line | Length (m) | Bearing | Latitude | Departure | |---|---|---|---|---| | AB | 178.600 | 127°30' | -108.725 | +141.693 | | BC | 228.400 | 48°45' | +150.595 | +171.720 | | DE | 126.700 | 258°30' | -25.260 | -124.156 | | **Sum** | | | +16.610 | +189.257 | The given bearings in whole circle form are AB = S52°30'E = 127°30', BC = N48°45'E = 48°45', CD = N18°15'W = 341°45', DE = S78°30'W = 258°30'. Let the unknown length of CD be $x$ (bearing 341°45'). Its latitude is $x\cos\theta$ and departure $x\sin\theta$. The line EA has length 238.80 m and must close the traverse, so $$ (16.610+x\cos\theta)^2+(189.257+x\sin\theta)^2=238.80^2 $$ with $\cos\theta=0.94970$ and $\sin\theta=(-0.31316)$ for 341°45'. #### Step 2: Solve for the length $$ x^2+2(16.610\times0.94970+189.257\times(-0.31316))x+(16.610^2+189.257^2-238.80^2)=0 $$ $$ x^2+(-86.988)x+(-20931.47)=0 $$ $$ x=\frac{-(-86.988)\pm\sqrt{91292.8}}{2}=194.567\ \text{or}\ -107.580 $$ The negative root has no meaning, so **length of CD = 194.567 m**. #### Step 3: Bearing of EA Latitude of EA $=-(16.610+194.567\times0.94970)=(-201.390)$ m and departure $=-(189.257+194.567\times(-0.31316))=(-128.325)$ m. $$ \tan\theta=\frac{(-128.325)}{(-201.390)}\ \Rightarrow\ \theta=212^\circ 30' $$ The signs of latitude and departure show the line lies in the south-west quadrant, so the whole circle bearing is **212°30'** (S 32°30' W). #### Step 4: Check | Line | Length (m) | Bearing | Latitude | Departure | |---|---|---|---|---| | AB | 178.600 | 127°30' | -108.725 | +141.693 | | BC | 228.400 | 48°45' | +150.595 | +171.720 | | CD | 194.567 | 341°45' | +184.781 | -60.931 | | DE | 126.700 | 258°30' | -25.260 | -124.156 | | EA | 238.800 | 212°30' | -201.390 | -128.325 | | **Sum** | | | -0.000 | +0.000 | Sums are nil, so the traverse closes. **Answer:** length of CD = 194.567 m; bearing of EA = 212°30'. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Length of CD and bearing of EA are unknown (EA's length is known). How are they found? A. Using SigmaLat = 0 and SigmaDep = 0, giving a quadratic in the length of CD B. Using the sine rule only C. Using Bowditch's rule D. By assuming CD = EA Answer: A. Two unknowns and two closure equations; the known EA length gives a quadratic for x. (Key point: Equations used) 2. What is S52°30'E in whole circle bearing? A. 52°30' B. 232°30' C. 127°30' D. 307°30' Answer: C. South-east quadrant: WCB = 180° - 52°30' = 127°30'. (Key point: Whole circle bearing) 3. The quadratic gives x = 194.567 or -107.580. Which is taken? A. -107.580 m, since it is smaller B. 194.567 m, since a length cannot be negative C. Their average D. Both, giving two lines Answer: B. A negative length has no meaning, so the positive root is accepted. (Key point: Choice of root) 4. EA has latitude -201.390 and departure -128.325. What is its whole circle bearing? A. 32°30', in the north-east quadrant B. 147°30', in the south-east quadrant C. 327°30', in the north-west quadrant D. 212°30', in the south-west quadrant Answer: D. Both negative means south-west: tan^-1(128.325/201.390) = 32°30' from south, so 180° + 32°30' = 212°30'. (Key point: Quadrant of EA) ## 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 - [Missing bearings of BC and DE in traverse](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-missing-bearings-bc-de) — Surveying II, chapter 1 - [Bearings of DE and EA in traverse ABCDE](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-abcde-bearings-de-ea) — Surveying II, chapter 1, asked 2× - [Bearings of BC and CD in traverse ABCDE](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-abcde-bearings-bc-cd) — Surveying II, chapter 1, asked 2× - [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 - [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 ## Related in other subjects - [Local attraction, bearings AB 305°00′ (EA 45°)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-ea-305-00) — Surveying I, chapter 4, asked 2× - [Local attraction: AB 69°30′ compass traverse](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-ab-69-30-reordered) — Surveying I, chapter 4 - [Local attraction, BC 79°30′ and DE 82°15′](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-ab-191-30-bc-79-30) — Surveying I, chapter 4 - [Local attraction, S 40°30′ W traverse](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-quadrantal-s40-30w) — Surveying I, chapter 4