Math / Geometry

QuestionPROBLEM 20: The azimuth of the sides of a traverse ABCDEF are given below. Compute the internal angles. Bearing of $A B=290^{\circ} 45^{\circ}$ Bearing of $B C=250^{\circ}+8^{\prime}$ Bearing of $C D=196^{\circ} 12^{\prime}$ Bearing of $D E=175^{\circ} 24^{\prime}$ Bearing of $E F=112^{\circ} 18^{\prime}$ Bearing of $F A=30^{\circ} 00^{\prime}$ Solution:

Studdy Solution

STEP 1

1. The traverse is a closed polygon.
2. Bearings are given in azimuths measured clockwise from the north.
3. Internal angles are computed by considering the change in direction between consecutive bearings.

STEP 2

1. Convert all bearings to decimal degrees.
2. Calculate the change in direction between consecutive bearings.
3. Compute the internal angles using the change in direction.

STEP 3

Convert the bearings from degrees and minutes to decimal degrees. The conversion formula is:
$\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60}$
For each bearing:
- Bearing of $AB = 290^\circ 45'$ $= 290 + \frac{45}{60} = 290.75^\circ$
- Bearing of $BC = 250^\circ 8'$ $= 250 + \frac{8}{60} = 250.1333^\circ$
- Bearing of $CD = 196^\circ 12'$ $= 196 + \frac{12}{60} = 196.2^\circ$
- Bearing of $DE = 175^\circ 24'$ $= 175 + \frac{24}{60} = 175.4^\circ$
- Bearing of $EF = 112^\circ 18'$ $= 112 + \frac{18}{60} = 112.3^\circ$
- Bearing of $FA = 30^\circ 00'$ $= 30.0^\circ$

STEP 4

Calculate the change in direction between consecutive bearings. The change in direction is the difference between two consecutive bearings, adjusted to ensure it is a positive angle less than $360^\circ$ .
- Change from $AB$ to $BC$ : $\Delta_1 = 250.1333^\circ - 290.75^\circ = -40.6167^\circ$ Adjust to positive by adding $360^\circ$ : $\Delta_1 = 360^\circ - 40.6167^\circ = 319.3833^\circ$
- Change from $BC$ to $CD$ : $\Delta_2 = 196.2^\circ - 250.1333^\circ = -53.9333^\circ$ Adjust to positive by adding $360^\circ$ : $\Delta_2 = 360^\circ - 53.9333^\circ = 306.0667^\circ$
- Change from $CD$ to $DE$ : $\Delta_3 = 175.4^\circ - 196.2^\circ = -20.8^\circ$ Adjust to positive by adding $360^\circ$ : $\Delta_3 = 360^\circ - 20.8^\circ = 339.2^\circ$
- Change from $DE$ to $EF$ : $\Delta_4 = 112.3^\circ - 175.4^\circ = -63.1^\circ$ Adjust to positive by adding $360^\circ$ : $\Delta_4 = 360^\circ - 63.1^\circ = 296.9^\circ$
- Change from $EF$ to $FA$ : $\Delta_5 = 30.0^\circ - 112.3^\circ = -82.3^\circ$ Adjust to positive by adding $360^\circ$ : $\Delta_5 = 360^\circ - 82.3^\circ = 277.7^\circ$
- Change from $FA$ to $AB$ : $\Delta_6 = 290.75^\circ - 30.0^\circ = 260.75^\circ$

STEP 5

Compute the internal angles using the change in direction. The internal angle at each vertex is given by:
$\text{Internal Angle} = 360^\circ - \Delta$
- Internal angle at $B$ : $\theta_B = 360^\circ - 319.3833^\circ = 40.6167^\circ$
- Internal angle at $C$ : $\theta_C = 360^\circ - 306.0667^\circ = 53.9333^\circ$
- Internal angle at $D$ : $\theta_D = 360^\circ - 339.2^\circ = 20.8^\circ$
- Internal angle at $E$ : $\theta_E = 360^\circ - 296.9^\circ = 63.1^\circ$
- Internal angle at $F$ : $\theta_F = 360^\circ - 277.7^\circ = 82.3^\circ$
- Internal angle at $A$ : $\theta_A = 360^\circ - 260.75^\circ = 99.25^\circ$
The internal angles of the traverse are approximately:
$\theta_B = 40.62^\circ, \quad \theta_C = 53.93^\circ, \quad \theta_D = 20.8^\circ, \quad \theta_E = 63.1^\circ, \quad \theta_F = 82.3^\circ, \quad \theta_A = 99.25^\circ$

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