### Bearing and Distance

The bearing and distance is an example of applied knowledge of trigonometry. The navigation used on land, sea or air was done using bearing (at least before GPS was invented and widely used). In this topic, we will learn how to use the bearing to calculate the distance or position of one place to another.

We will use the convention for bearing as follows:

– three-figure or three-digit

– North or N is the reference for \({\small 0^{\large{\circ}}}\)

– measured in the clockwise direction from the North

I have put together some of the questions I received in the comment section below. You can try these questions also to further your understanding on this topic.

To check your answer, you can look through the solutions that I have posted either in Youtube videos or Instagram posts.

You can subscribe, like or follow my youtube channel and IG account. I will keep updating my IG daily post, preferably.

Furthermore, you can find some examples and more practices below! =).

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QUESTIONS FROM STUDENTS:

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\({\small 2.\enspace}\) A boat sails round a quadrangular course

*ABCD*, starting from

*A*to

*B*in 4 km due east of

*A*,

*C*is 3 km due south of

*B*and

*D*is 4 km S50W from

*C*.

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What is the distance and bearing of

*A*from

*D*?

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\({\small 3.\enspace}\) \({\small\hspace{0.2em}\left(\textrm{a}\right).\hspace{0.8em}}\) A village is 10 km on a bearing \({\small 050^{\large{\circ}}}\) from a point

*O*. How far is the village North of

*O*?

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\({\small\hspace{1.6em}\left(\textrm{b}\right).\hspace{0.8em}}\) The angle of elevation of the top of a building from a point 80 m away on a level ground is \({\small 25^{\large{\circ}}}\).

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Calculate the height of the building to the nearest metre.

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\({\small 4.\enspace}\) Town

*F*is 50 km east of Town

*G*. Town

*H*is on a bearing of \({\small 040^{\large{\circ}}}\) from Town

*F*. The distance from

*F*to

*H*is 65 km.

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Calculate, to the nearest km, the actual distance

*GH*.

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Calculate, to the nearest degree, the bearing of

*H*from

*G*.

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\({\small 5.\enspace}\) A village R is 10 km from a point P on a bearing \({\small 025^{\large{\circ}}}\) from P. Another village A is 6 km from P on a bearing of \({\small 016^{\large{\circ}}}\). Calculate,

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\({\small\hspace{1.2em}\left(\textrm{a}\right).\hspace{0.8em}}\) the distance of R from A and

\({\small\hspace{1.2em}\left(\textrm{b}\right).\hspace{0.8em}}\) the bearing of R from A

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\({\small 6.\enspace}\) The bearing of city

*R*from city

*M*is \({\small 058^{\large{\circ}}}\) and the bearing of city

*K*from city

*M*is \({\small 310^{\large{\circ}}}\).

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If the distance from

*M*to

*R*is 40 km and that of

*M*to

*K*is 70 km,

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What is the distance between

*K*and

*R*to 1 decimal place and bearing of

*K*from

*R*

to the nearest degree?

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\({\small 7.\enspace}\) The bearing of two points

*Q*and

*R*from a point

*P*are \({\small 050^{\large{\circ}}}\) and \({\small 120^{\large{\circ}}}\) respectively.

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If

*PQ*= 12 cm and

*PR*= 5 cm, find the distance

*QR*.

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\({\small 8.\enspace}\) A ship leaves a port at noon and has a bearing of \({\small 207^{\large{\circ}}}\). If the ship is traveling at 20 miles per hour, how many miles south and how many miles west is the ship from its departure point at 6 pm?

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\({\small 9.\enspace}\) From a point

*H*at a harbour, the bearing of two ships

*A*and

*B*on the high sea are \({\small 160^{\large{\circ}}}\) and \({\small 220^{\large{\circ}}}\) respectively.

*B*is 14 km from

*H*and the bearing of

*A*from

*B*is \({\small 085^{\large{\circ}}}\).

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\({\small\hspace{1.2em}\left(\textrm{a}\right).\hspace{0.8em}}\) How far apart are the two ships?

\({\small\hspace{1.2em}\left(\textrm{b}\right).\hspace{0.8em}}\) What is the bearing of

*B*from

*A*?

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\({\small 10.\enspace}\) An aircraft travels 120 km on a bearing of \({\small 062^{\large{\circ}}}\) and then 95 km on a bearing of \({\small 305^{\large{\circ}}}\). How far is the aircraft from the starting point?

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\({\small 11.\enspace}\) Two boats

*A*and

*B*left a port

*C*at the same time on different routes.

*B*travelled on a bearing of \({\small 150^{\large{\circ}}}\) and

*A*travelled on north side of

*B*. When

*A*had travelled 8 km and

*B*travelled 10 km, the distance between the two boats was found to be 12 km. Calculate the bearing of

*A*’s route from

*C*.

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\({\small 12.\enspace}\) An aeroplane flies from town

*X*on a bearing of \({\small 45^{\large{\circ}}}\)E to another town

*Y*with a distance of 200 km. It then changes course and flies to another town

*Z*on a bearing of S\({\small 60^{\large{\circ}}}\)E. If

*Z*is directly east of

*X*, calculate correct to 3 s.f.:

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\({\small\hspace{1.2em}\left(\textrm{a}\right).\hspace{0.8em}}\) the distance from

*X*to

*Z*.

\({\small\hspace{1.2em}\left(\textrm{b}\right).\hspace{0.8em}}\) the distance from

*Y*to

*XZ*.

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\({\small 13.\enspace}\)

*A*,

*B*,

*C*are three locations on the same horizontal plane.

*B*is on a bearing of \({\small 041^{\large{\circ}}}\) from

*A*and the distance is 40 km.

*C*is directly due north of

*A*and the distance between

*B*and

*C*is 35 km. Find the bearing of

*C*from

*B*.

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\({\small 14.\enspace}\) The bearing of

*A*from

*B*is \({\small 129^{\large{\circ}}}\) and the bearing of

*C*from

*B*is \({\small 219^{\large{\circ}}}\). If

*B*is equidistant from

*A*and

*C*, find the bearing of

*C*from

*A*.

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\({\small 14.\enspace}\)

*A*,

*B*,

*C*are three locations on the same horizontal plane.

*B*is on a bearing of \({\small 041^{\large{\circ}}}\) from

*A*and the distance is 40 km.

*C*is directly due north of

*A*and the distance between

*B*and

*C*is 35 km. Find the bearing of

*C*from

*B*.

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\({\small 15.\enspace}\) A ship sails 50 km north from a point

*A*, then 25 km west and finally 50 km on a bearing of \({\small 315^{\large{\circ}}}\).

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\({\small\hspace{1.2em}\left(\textrm{a}\right).\hspace{0.8em}}\) How far west is the ship from the point

*A*?

\({\small\hspace{1.2em}\left(\textrm{b}\right).\hspace{0.8em}}\) How far north is the ship from the point

*A*?

\({\small\hspace{1.2em}\left(\textrm{c}\right).\hspace{0.8em}}\) Find the distance and bearing of the ship destination from

*A*?

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EXAMPLE:

*A, B*and

*C*are three towns located on a horizontal ground. It is given that

*AC*\(=\) 22 km and

*BC*\(=\) 18 km.

*C*is at a bearing of \({\small 025^{\large{\circ}}}\) from

*A*and \({\small 337^{\large{\circ}}}\) from

*B.*

\({\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}\) Show that \({\small \angle {ACB} = 48^{\large{\circ}}}\).

\({\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}\) Calculate

\({\small\hspace{2.8em}\left(i\right).\hspace{0.7em}}\) the distance

*AB,*

\({\small\hspace{2.8em}\left(ii\right).\hspace{0.7em}}\) the bearing of

*A*from

*B,*

\({\small\hspace{2.8em}\left(iii\right).\hspace{0.5em}}\) the area of triangle

*ABC,*

\({\small\hspace{2.8em}\left(iv\right).\hspace{0.7em}}\) the shortest distance from

*B*to

*AC.*

\({\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}\) A helicopter,

*H,*is hovering directly above a point

*D,*nearest to

*B*on

*AC*. If the angle of elevation of

*H*seen from

*B*is \({\small 10^{\large{\circ}}}\), calculate the height of

*H*above

*D.*

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\({\small 2.\enspace}\) A ship leaves port at noon and has a bearing of \({\small 207^{\large{\circ}}}\). If the ship is traveling at 20 miles per hour, how many miles south and how many miles west is the ship from its departure point at 6 pm?

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\({\small 3.\enspace}\) From the lookout tower

*A,*a column of smoke is sighted due south. From a second lookout tower

*B,*5 miles west of

*A,*the smoke is observed with a bearing of \({\small 117^{\large{\circ}}}\). How far is the fire from tower

*B*? How far is it from tower

*A*?

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\({\small 4.\enspace}\) Pittsburgh is 117 miles due south of Kansas City. A plane leaves Kansas City at noon and flies 200 mph with a bearing of \({\small 201^{\large{\circ}}}\). When, to the nearest minute, will the plane be due west of Pittsburgh?

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\({\small 5.\enspace}\) Two ships

*A*and

*B*leave port at the same time with ship

*A*sailing with a bearing of \({\small 023^{\large{\circ}}}\) at the speed of 11 mph and ship

*B*sailing with a bearing of \({\small 113^{\large{\circ}}}\) at 15 mph. Approximate the bearing from ship

*B*to ship

*A*one hour later.

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PRACTICE MORE WITH THESE QUESTIONS BELOW!

\({\small 1.\enspace}\) Two ships, *A* and *B* leave port at 13 00 hours. Ship *A* travels at a constant speed of 18 km per hour on a bearing of \({\small 070^{\large{\circ}}}\). Ship *B* travels at a constant speed of 25 km per hour on a bearing of \({\small 152^{\large{\circ}}}\). Calculate the distance between *A* and *B* at 15 00 hours.

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\({\small 2. \enspace}\) A plane is 120 miles north and 85 miles east of an airport. If the pilot wants to fly directly to the airport, what bearing should be taken?

\({\small 3. \enspace}\) Ohio is 190 miles due north of Virginia. Stamford is due east of Ohio and is 460 miles from Virginia. What is the bearing angle heading from Virginia to Stamford? What is the bearing angle from Stamford to Virginia?

\({\small 4. \enspace}\) *D* is 30 km due east of *A*. A ship sails from *A* on a certain bearing for 15 km to *B*. It then sails to *C*, which is 40 km from *A* and due north of *D*. Finally it sails due south to *D*. *AB* is the bisector of \({\small \angle {EAC}}\). Calculate the total distance for the whole journey.

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\({\small 5. \enspace}\) An observer at a radar station plots an aircraft due north of him and at a range of 50 km, measured horizontally. The aircraft keeps on traveling due west at a constant height. The bearing of the plane from the observer 10 minutes later is \({\small 300^{\large{\circ}}}\). Calculate the speed of the aircraft in km/h.

\({\small 6. \enspace}\) *N* is north of *S*. *NS* is the tangent to the circle, radius 1 km, at *T*. A man walks 2 km along the circle from *T* to *P*. What is the distance and bearing of *P* from *T*?

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\({\small 7. \enspace}\) In the diagram, *A*, *B*, *C* and *D* are four points on a horizontal field. *A* is north of *B*. The bearing of *D* from *A* is \({\small 115^{\large{\circ}}}\) and the bearing of *D* from *B* is \({\small 25^{\large{\circ}}}\). \({\small \angle {BDC} = 130^{\large{\circ}}}\), *BD* \(=\) 150 m and *CD* \(=\) 110 m.

\({\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}\) Find the value of \({\small \angle {ADB}}\).

\({\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}\) Find the bearing of *B* from *D*.

\({\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}\) Find the bearing of *C* from *D*.

\({\small\hspace{1.2em}\left(d\right).\hspace{0.8em}}\) Calculate the shortest distance of point *A* to line *DB*.

\({\small\hspace{1.2em}\left(e\right).\hspace{0.8em}}\) Calculate the length of *BC*.

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\({\small 8. \enspace}\) In the diagram (not drawn to scale), *E*, *G* and *C* represents three points at the Esplanade, Gardens by the Bay and the City Hall respectively. Given that *EG* \(=\) 1125 m and *CE* \(=\) 957 m, the bearing of *G* from *E* is \({\small 135^{\large{\circ}}}\) and \({\small \angle {CEG} = 60^{\large{\circ}}}\). Find

\({\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}\) the bearing of *E* from *C*,

\({\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}\) the distance between *C* and *G*,

\({\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}\) \({\small \angle {EGC}}\),

\({\small\hspace{1.2em}\left(d\right).\hspace{0.8em}}\) the bearing of *C* from *G*.

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\({\small 9. \enspace}\) In the diagram, *Q* is due east of *P*. Given that *PQ* \(=\) 55 m and *RQ* \(=\) 32 m and \({\small \angle {PRQ} = 105^{\large{\circ}}}\). Find

\({\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}\) The bearing of *R* from *P*,

\({\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}\) The area of \({\small \triangle {PRQ}}\).

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\({\small 10.\enspace}\) In the diagram, *A*, *B*, *C* and *D* are four corners of a field. *AD* \(=\) 53 m, *DC* \(=\) 87 m, *BC* \(=\) 66 m, \({\small \angle {ABC} = 90^{\large{\circ}}}\) and \({\small \angle {ACB} = 58^{\large{\circ}}}\).

\({\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}\) Calculate *AC*.

\({\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}\) The bearing of *B* from *C* is \({\small 340^{\large{\circ}}}\). Calculate

\({\small\hspace{2.8em}\left(i\right).\hspace{0.7em}}\) the bearing of *A* from *C*,

\({\small\hspace{2.8em}\left(ii\right).\hspace{0.7em}}\) the bearing of *A* from *B*.

\({\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}\) Calculate

\({\small\hspace{2.8em}\left(i\right).\hspace{0.7em}}\) the angle of \({\small \angle {ADC}}\),

\({\small\hspace{2.8em}\left(ii\right).\hspace{0.7em}}\) the area of \({\small \triangle {ADC}}\),

\({\small\hspace{2.8em}\left(iii\right).\hspace{0.5em}}\) the shortest distance from *A* to *DC.*

\({\small\hspace{1.2em}\left(d\right).\hspace{0.8em}}\) An eagle is hovering vertically above *A*. The angle of elevation of the eagle from *D* is \({\small 42^{\large{\circ}}}\). Calculate the greatest angle of elevation of the eagle from a point on *DC.*

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As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .