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To calculate the coordinates of points B and C, we use the formulas for coordinate geometry:

Mathematics

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2 parts
a

(530.317 m E, 601.780 m N)\text{(530.317 m E, 601.780 m N)}

b

(468.382 m E, 465.741 m N)\text{(468.382 m E, 465.741 m N)}

Step-by-step solution

6 steps

To calculate the coordinates of points B and C, we use the formulas for coordinate geometry:

EX=EA+DAXsin⁡(θAX)E_X = E_A + D_{AX} \sin(\theta_{AX}) NX=NA+DAXcos⁡(θAX)N_X = N_A + D_{AX} \cos(\theta_{AX})

where (EA,NA)(E_A, N_A) are the coordinates of point A, DAXD_{AX} is the distance from A to X, and θAX\theta_{AX} is the azimuth from A to X.

Given:
Point A coordinates: (EA,NA)=(500 m E,550 m N)(E_A, N_A) = (500 \text{ m E}, 550 \text{ m N})

Part a· 3 steps

Calculate coordinates for point B

Distance DAB=60 mD_{AB} = 60 \text{ m}
Azimuth θAB=30∘20′20′′\theta_{AB} = 30^\circ 20' 20''

  1. Convert the azimuth to decimal degrees.

    θAB=30∘+2060∘+203600∘=30∘+0.333333∘+0.005556∘=30.338889∘\theta_{AB} = 30^\circ + \frac{20}{60}^\circ + \frac{20}{3600}^\circ = 30^\circ + 0.333333^\circ + 0.005556^\circ = 30.338889^\circ

  2. Calculate the Easting coordinate for point B (EBE_B).

    EB=EA+DABsin⁡(θAB)E_B = E_A + D_{AB} \sin(\theta_{AB}) EB=500 m+60 m×sin⁡(30.338889∘)E_B = 500 \text{ m} + 60 \text{ m} \times \sin(30.338889^\circ) EB=500 m+60 m×0.505291E_B = 500 \text{ m} + 60 \text{ m} \times 0.505291 EB=500 m+30.31746 mE_B = 500 \text{ m} + 30.31746 \text{ m} EB=530.317 mE_B = 530.317 \text{ m}

  3. Calculate the Northing coordinate for point B (NBN_B).

    NB=NA+DABcos⁡(θAB)N_B = N_A + D_{AB} \cos(\theta_{AB}) NB=550 m+60 m×cos⁡(30.338889∘)N_B = 550 \text{ m} + 60 \text{ m} \times \cos(30.338889^\circ) NB=550 m+60 m×0.863005N_B = 550 \text{ m} + 60 \text{ m} \times 0.863005 NB=550 m+51.78030 mN_B = 550 \text{ m} + 51.78030 \text{ m} NB=601.780 mN_B = 601.780 \text{ m}

    The coordinates of point B are (530.317 m E, 601.780 m N)\boxed{\text{(530.317 m E, 601.780 m N)}}.

Part b

(468.382 m E, 465.741 m N)\text{(468.382 m E, 465.741 m N)}

Calculate coordinates for point C

Distance DAC=90 mD_{AC} = 90 \text{ m}
Azimuth θAC=200∘33′55′′\theta_{AC} = 200^\circ 33' 55''

  1. Convert the azimuth to decimal degrees.

    θAC=200∘+3360∘+553600∘=200∘+0.55∘+0.015278∘=200.565278∘\theta_{AC} = 200^\circ + \frac{33}{60}^\circ + \frac{55}{3600}^\circ = 200^\circ + 0.55^\circ + 0.015278^\circ = 200.565278^\circ

  2. Calculate the Easting coordinate for point C (ECE_C).

    EC=EA+DACsin⁡(θAC)E_C = E_A + D_{AC} \sin(\theta_{AC}) EC=500 m+90 m×sin⁡(200.565278∘)E_C = 500 \text{ m} + 90 \text{ m} \times \sin(200.565278^\circ) EC=500 m+90 m×(−0.351307)E_C = 500 \text{ m} + 90 \text{ m} \times (-0.351307) EC=500 m−31.61763 mE_C = 500 \text{ m} - 31.61763 \text{ m} EC=468.382 mE_C = 468.382 \text{ m}

  3. Calculate the Northing coordinate for point C (NCN_C).

    NC=NA+DACcos⁡(θAC)N_C = N_A + D_{AC} \cos(\theta_{AC}) NC=550 m+90 m×cos⁡(200.565278∘)N_C = 550 \text{ m} + 90 \text{ m} \times \cos(200.565278^\circ) NC=550 m+90 m×(−0.936216)N_C = 550 \text{ m} + 90 \text{ m} \times (-0.936216) NC=550 m−84.25944 mN_C = 550 \text{ m} - 84.25944 \text{ m} NC=465.741 mN_C = 465.741 \text{ m}

    The coordinates of point C are (468.382 m E, 465.741 m N)\boxed{\text{(468.382 m E, 465.741 m N)}}.

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