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Corrected and true bearings, 8°40′E

Short answer

Corrected fore bearings: AB 66°15', BC 132°00', CD 217°30', DA 306°45'. True bearings (magnetic +8°40'): AB 74°55', BC 140°40', CD 226°10', DA 315°25'.

The following bearings were observed in running a compass traverse.

LineFore bearingBack bearing
AB66°15′244°0′
BC129°45′313°0′
CD218°30′37°30′
DA306°45′126°45′

Find the corrected fore and back bearings, and the true bearings of the lines, given that the magnetic declination is 8°40′E.

Line DA has FB − BB = 180°00′, so stations D and A are free; B and C are suspected. Declination is 8°40′ East, so true bearing = magnetic bearing + 8°40′.

Local attraction at a station is detected by comparing the fore bearing (FB) and back bearing (BB) of each line: if there is no local attraction at either end, FB−BB=180∘\text{FB} - \text{BB} = 180^\circ exactly.

Step 1: Check each line

LineFBBBFB − BBRemark
AB66°15′244°00′182°15′Differs by +2°15′
BC129°45′313°00′176°45′Differs by −3°15′
CD218°30′37°30′181°00′Differs by +1°00′
DA306°45′126°45′180°00′Free, difference = 180°

Line DA has a difference of exactly 180°, so stations D and A are free from local attraction. The other stations are corrected one after another, starting from A.

Step 2: Correct the stations in turn

  • Line AB: FB at A = 66°15′ is correct. The correct BB is 246°15′, observed 244°00′, so the correction at B is +2°15′.
  • Line BC: FB at B = 129°45′ with correction +2°15′ gives 132°00′. The correct BB is 312°00′, observed 313°00′, so the correction at C is −1°00′.
  • Line CD (check): FB at C = 218°30′ with correction −1°00′ gives 217°30′. The correct BB is 37°30′, which equals the observed value, so D needs no correction, as expected.
StationABCD
Correction0+2°15′−1°00′0

Stations affected by local attraction: B, C.

Step 3: Corrected bearings

LineObserved FBCorrectionCorrected FBObserved BBCorrected BBTrue FBTrue BB
AB66°15′066°15′244°00′246°15′74°55′254°55′
BC129°45′+2°15′132°00′313°00′312°00′140°40′320°40′
CD218°30′−1°00′217°30′37°30′37°30′226°10′46°10′
DA306°45′0306°45′126°45′126°45′315°25′135°25′

Check: corrected FB − corrected BB = 180° for every line.

Answer: Corrected bearings: AB = 66°15′ / 246°15′, BC = 132°00′ / 312°00′, CD = 217°30′ / 37°30′, DA = 306°45′ / 126°45′ (FB / BB). True bearings (FB): AB = 74°55′, BC = 140°40′, CD = 226°10′, DA = 315°25′.

✓ Answer checked by Claude Opus 5.5 (high effort)

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