It is needed to establish the coordinates of a station A. Two stations B and C are available whose coordinates are given in the table. Length of line CA = 1212.38 m; \(\angle BCA = 64^\circ\). The Easting of station A is ______ m (Rounded off to two decimal places). Assume station A lies to the west of the traverse line BC.
Station
Easting (m)
Northing (m)
B
11054.09
9484.37
C
10827.62
10112.15
Show Hint
Find the bearing of BC from the coordinates, add or subtract the given angle at C to get the bearing of CA, then resolve the length CA along that bearing, choosing the direction that places A west of line BC.
Step 1: Find the direction of line CB from the given coordinates. The coordinate differences from C to B are $\Delta E_{CB} = E_B - E_C = 11054.09 - 10827.62 = 226.47$ m and $\Delta N_{CB} = N_B - N_C = 9484.37 - 10112.15 = -627.78$ m; the vector C to B moves east and south, so it lies in the south-east quadrant. Its angle from north is $\tan^{-1}(226.47/627.78) = 19.837^\circ \approx 19^\circ50'$, so the azimuth of CB is $180^\circ - 19^\circ50' = 160^\circ10'$. Step 2: Rotate this azimuth by the given angle to obtain the direction of CA. At C, the angle between CB and CA is $64^\circ$. Turning CB clockwise gives $160^\circ10' + 64^\circ = 224^\circ10'$; turning it counter-clockwise gives $160^\circ10' - 64^\circ = 96^\circ10'$. Station A must fall west of line BC. The clockwise rotation ($224^\circ10'$, a south-westerly direction from C) carries the point to the west side of BC, while the counter-clockwise one ($96^\circ10'$) carries it to the east side. So the azimuth of CA is $224^\circ10'$. Step 3: Resolve the length CA along this azimuth. With $CA = 1212.38$ m: \[ \Delta E = CA\sin(224^\circ10'), \quad \Delta N = CA\cos(224^\circ10') \] Since $224^\circ10' = 180^\circ+44^\circ10'$, $\sin(224^\circ10') = -\sin(44^\circ10') = -0.6967$ and $\cos(224^\circ10') = -\cos(44^\circ10') = -0.7173$. So \[ \Delta E = 1212.38 \times (-0.6967) = -844.67 \text{ m}, \quad \Delta N = 1212.38\times(-0.7173) = -869.68 \text{ m} \] Step 4: Add the components to the coordinates of C. \[ E_A = 10827.62 + (-844.67) = 9982.95 \text{ m} \] (and $N_A = 10112.15 - 869.68 = 9242.47$ m, though only the easting is asked here.) Step 5: Final value. \[ \boxed{E_A \approx 9982.95 \text{ m}} \]
