--- title: "Omitted quantities in closed traverse ABCDE" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:omitted-quantities-closed-traverse-abcde" source: https://www.bctengineer.com/q/semester_4/surveying_ii/omitted-quantities-closed-traverse-abcde site: BCT Engineering --- # Omitted quantities in closed traverse ABCDE Surveying II · Chapter 1: Traversing - Asked in: 2074 Bhadra, 10 marks - Video: [The length and bearings of a closed traverse PQRSP are given in the table.The missing length and](https://www.youtube.com/watch?v=CkQNnP2pZh4) — Unique Series - Civil Mantraa - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/omitted-quantities-closed-traverse-abcde - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question Calculate the omitted quantities in the closed traverse ABCDE given below. | Line | Length | Bearing | |---|---|---| | AB | 282.20 | 61°30' | | BC | ? | 151°24' | | CD | 324.70 | 201°02' | | DE | 381.60 | 280°14' | | EA | 359.60 | ? | ## Short answer Length BC = 274.321 m, bearing EA = 18°16'. Solved by setting ΣLat=0, ΣDep=0: unknown BC length from quadratic closure equation, then bearing EA computed from resultant latitude and departure. ## Answer For a closed traverse $\Sigma\text{Lat}=0$ and $\Sigma\text{Dep}=0$. Two quantities are unknown, the length of BC 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 | 282.200 | 61°30' | +134.654 | +248.002 | | CD | 324.700 | 201°02' | -303.066 | -116.538 | | DE | 381.600 | 280°14' | +67.794 | -375.530 | | **Sum** | | | -100.618 | -244.066 | Let the unknown length of BC be $x$ (bearing 151°24'). Its latitude is $x\cos\theta$ and departure $x\sin\theta$. The line EA has length 359.60 m and must close the traverse, so $$ ((-100.618)+x\cos\theta)^2+((-244.066)+x\sin\theta)^2=359.60^2 $$ with $\cos\theta=(-0.87798)$ and $\sin\theta=0.47869$ for 151°24'. #### Step 2: Solve for the length $$ x^2+2((-100.618)\times(-0.87798)+(-244.066)\times0.47869)x+((-100.618)^2+(-244.066)^2-359.60^2)=0 $$ $$ x^2+(-56.984)x+(-59620.10)=0 $$ $$ x=\frac{-(-56.984)\pm\sqrt{241727.5}}{2}=274.321\ \text{or}\ -217.337 $$ The negative root has no meaning, so **length of BC = 274.321 m**. #### Step 3: Bearing of EA Latitude of EA $=-((-100.618)+274.321\times(-0.87798))=341.467$ m and departure $=-((-244.066)+274.321\times0.47869)=112.751$ m. $$ \tan\theta=\frac{112.751}{341.467}\ \Rightarrow\ \theta=18^\circ 16' $$ The signs of latitude and departure show the line lies in the north-east quadrant, so the whole circle bearing is **18°16'** (N 18°16' E). #### Step 4: Check | Line | Length (m) | Bearing | Latitude | Departure | |---|---|---|---|---| | AB | 282.200 | 61°30' | +134.654 | +248.002 | | BC | 274.321 | 151°24' | -240.849 | +131.315 | | CD | 324.700 | 201°02' | -303.066 | -116.538 | | DE | 381.600 | 280°14' | +67.794 | -375.530 | | EA | 359.600 | 18°16' | +341.467 | +112.751 | | **Sum** | | | +0.000 | +0.000 | Sums are nil, so the traverse closes. **Answer:** length of BC = 274.321 m; bearing of EA = 18°16'. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Length of BC and bearing of EA are unknown. How are they found? A. From Bowditch's rule B. From SigmaLat = 0 and SigmaDep = 0, using EA's length to form a quadratic in BC's length C. From the sine rule with angles at A and B D. By measuring on a plot only Answer: B. Two closure equations for two unknowns. (Key point: Method) 2. The roots for BC are 274.321 and -217.337. Which is correct? A. 274.321 m, because length is positive B. -217.337 m, as it is smaller in size C. Their mean D. Neither Answer: A. Negative length is meaningless. (Key point: Choice of root) 3. How are latitude and departure of EA found? A. Equal to the sum of the others B. Equal to BC's values C. Equal and opposite to the sum of latitudes and departures of all the other lines D. Found from the average bearing Answer: C. Closure requires the sums to be zero, so EA cancels the others (+341.467, +112.751). (Key point: EA components) 4. EA has positive latitude and departure (341.467, 112.751). What is its bearing? A. 161°44', in the south-east quadrant B. 198°16', in the south-west quadrant C. 341°44', in the north-west quadrant D. 18°16', in the north-east quadrant Answer: D. tan^-1(112.751/341.467) = 18°16' and both positive means NE. 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