- 2067 Asar · 3+5 marks
Direction theodolite, least count 6″, station D
Short answer
Mean direction from D: to C = 0°00′00″, to B = 83°06′48″, to A = 242°35′24″. Angles from differences: CDB = 83°06′48″, BDA = 159°28′36″, ADC = 117°24′36″; sum 360°, horizon closed.
A direction theodolite with a least count of 6 seconds is set over station D to measure directions to stations C, B and A. The observed directions from that position are as follows:
| Stations | Telescope | Horizontal Circle Readings |
|---|---|---|
| C | Direct | 00°00′00″ |
| Reversed | 179°59′48″ | |
| B | Direct | 83°06′48″ |
| Reversed | 263°06′36″ | |
| A | Direct | 242°35′24″ |
| Reversed | 62°35′12″ |
Compute an abstract of average direction of CD, CB and CA and compute the average angles of CDB, BDA and ADC.
Method. A direction theodolite (least count 6″) reads directions on a fixed circle. At D, the directions to C, B and A are observed on both faces. The mean direction of each = [Direct + (Reversed - 180°)]/2. All directions are then referred to C (the reference object) by subtracting the mean direction of C, which gives the abstract of average directions. The angles are the differences between the directions.
Abstract of average directions (at D)
| Station | Telescope | Circle reading | Reading ∓ 180° | Mean direction | Direction relative to C (C = 0°) |
|---|---|---|---|---|---|
| C | Direct | 00°00′00″ | |||
| Reversed | 179°59′48″ | 359°59′48″ | 359°59′54″ | 0°00′00″ | |
| B | Direct | 83°06′48″ | |||
| Reversed | 263°06′36″ | 83°06′36″ | 83°06′42″ | 83°06′48″ | |
| A | Direct | 242°35′24″ | |||
| Reversed | 62°35′12″ | 242°35′12″ | 242°35′18″ | 242°35′24″ |
The reduced average directions are: DC = 0°00′00″, DB = 83°06′48″, DA = 242°35′24″.
Average angles
Check: 83°06′48″ + 159°28′36″ + 117°24′36″ = 360°00′00″, so the horizon closes.
Answer: average directions DC = 0°00′00″, DB = 83°06′48″, DA = 242°35′24″; average angles CDB = 83°06′48″, BDA = 159°28′36″, ADC = 117°24′36″.
✓ Answer checked by Claude Opus 5.5 (high effort)
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