### 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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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?

\(\\[1pt]\)

\({\small 5.\enspace}\) 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 =) .

1cm to 20km is used find the distances and bearings of p,q,r,s and t from o

1cm to 20km is used , find the distances and bearings of p,q,r,s and t from o

An aeroplane flies for 4hrs from position P on a bearing of North 60 degree west to a point Q at an average speed of 60km/h.

The aeroplane then flies on a bearing of 150 degree Q to another point R 500km away. Calculate correct to 3 significant figures and show the diagram.

( a) the distance PR

(b) the bearing of R from P

A man start from a point x and walks 285m to y on a bearing of 078. He then walks due south to a point z which is 307m from x. Illustrate the information on a diagram. Find the bearing of x and z and find the distance between y and z.

A car travels from town P for 50km due North to town Q. It then travels 80km on a bearing of 270° to town R.

Calculate the shortest distance from town P to town R.

The bearing of A from B is 192 degrees. What is the bearing of B from A?

Two boats leave a port at the same time. The first leaves at 15km/hr on a bearing 135° while the second travels at 20km/hr on a bearing 063°. If after 2hours, the second boat is directly north of the first boat. Calculate their distance apart.

Pls help me with this I don’t understand,a students works 50m on a bearing of 25degrees and then 200m due east,how far is she from her starting point

Weldon sir\ma please i need your help i need a teacher for this topic

Explain more on it

Weldon sir\ma please i need your help i need a teacher for this topic

Two boats A and B left a port C at the same tie on dfferent routes.B travelled on a baring of 150 and A travelled on north side of B. When A had travelled 8km and B travelled 10km, the distance between the btwo boats was found to be 12km. Calculate the bearing of A’s route from C

Check out the solution here, Jackson:

https://www.instagram.com/p/CPCfSNEBhek/

Cheers =)

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uptuition with mr will

An aircraft travels 120km on a bearing of 062° and then 95km on a bearing of 305°. How far is the aircraft from the starting point

Check out the solution here, Kopji:

https://www.instagram.com/p/CPCfGwHBodI/

Cheers =)

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An aircraft travels 120km on a bearing of 062 and 95km on a bearing of 305. How far is the aircraft from its starting point

Comment how do they solve this

Two men p and q set of a base camp R prospecting from oil.P moves 20km on a bearing 205 degree and Q moves 15km on bearing 060 degree to calculate the distance of Q from P and the bearing of Q and P

Comment how do they solve this

A navy ship leaves port A on a bearing of 120° and travels 200 kilometers to port B.it then travels 250 kilometers from port B to port C on a bearing of 225°.

¡)Draw a diagram that best represents the route of the ship.

¡¡)Calculate the direct distance from port A to port C to the nearest kilometer.

A man sets to travel from P to R via Q. From P he travels a distance off 12 kilometres on a bearing 30 degree to q. From q he then travels a further 5 km due east. I. How far R is North of p. I I how far r is East of p. III. Evaluate pr

an aeroplane flew from a city X on a b X earing 225° to another city Y a distance 200km away. it then flew from city Y or a bearing of N30°W to city Z a distance of 250km. calculate: (1) the distance from city X connect to the nearest whole number. (2) the bearing of city X from Z

Commenting iam in trouble its getting me mad

Please somebody,anybody answer this question for me. Three towns XYZ are such that the distance between X and Y is 15Km and the distance between X and Z is 9 Km.If the bearing of Y from X is 70°and the bearing of Z from X is 310°.Calculate;

(i)the distance between Y and Z

(ii)the bearing of Z from Y

(iii)the bearing of Y from Z

Three towns XYZ are such that the distance between X and Y is 15Km and the distance between X and Z is 9 Km.If the bearing of Y from X is 70°and the bearing of Z from X is 310°.Calculate;

(i)the distance between Y and Z

(ii)the bearing of Z from Y

(iii)the bearing of Y from Z

I don’t understand the 9th problem :(( How can it be solved?

9. In the diagram, Q is due east of P. Given that PQ = 55 m and RQ = 32 m and ∠PRQ=105∘. Find

(a). The bearing of R from P

How do I find a?

never mind. ive solved it already :))

Hello I need help

Town F is 50 km east of town G

Town H is on a bearing of 040 from Town F.

The distance from F to H is 65 km

Calculate ,to the nearest km, the actual distance GH .

Calculate, to the nearest degree, the bearing of H from G.

Check out the solution here, Asia:

https://www.instagram.com/p/CMU0DKEllAC

Cheers =)

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From a point H at a harbour,the bearing of two ship A and B,on the high sea are 160 and 220 respectively.B is 14km from H and bearing of A from B is 85

How far apart are the two ships.

What is the bearing of B from A

Check out the solution here, Precious:

https://www.instagram.com/p/CMgdBoclhWh/

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b) The bearing of a ship

R

from a port

P

is 280⁰ and its bearing from a town

T

is

010 .

If the town

T

lies due west of the port

P,

find the bearing of

(i) T from

R

A ship leaves a port at noon and has a bearing of 207 degree. If the ship is traveling at 20 miles per hour, how many miles south and how many miles west is the ship from it’s departure point at 6 pm?

Check out the solution here, Joshua:

https://www.instagram.com/p/CMgc_U1ls3P/

Cheers =)

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I don’t understand the work.

Please can u help me with this question. The bearing of two point Q and R from a point P are 050 and 120 respectively if [PQ]=12cm and [PR]=5cm,find the distance QR

Check out the solution here, Anon:

https://www.instagram.com/p/CMgc9WxlqxU/

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the bearing of city R from city M is 0.58degres and the bearing of city K from city M is 310degres if the distance from M to R is 40km and that of M to K is 70km what is the distance between K and R to 1 decimal place and bearing of K from R to the nearest degree

Check out the solution here, Iliyasu:

https://www.instagram.com/p/CMgc7IcFcdM/

Cheers =)

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A village R is 10km from a point P on a bearing 025 degrees from P. Another village A is 6km from P on a bearing 16 degree. Calculate

a) distance of R from A

b) bearing of R from A

Check out the solution here, Promise:

https://www.instagram.com/p/CMgc4Tnl78K/

Cheers =)

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This was really helpful thank you so much

Help me with this question.factorise 5b_3

A ship is 4km due north and then 3km on a bearing of 60°.find the distance from his original position.

You can take a look at my explanation below:

https://youtu.be/I2QdTbg_7-M

Cheers,

Mr Will

I need the solution

I need ur help

mohammed treks 3km from point A on a bearing of 023°. he then treks 4km on a bearing of 113°to a point B. what is the bearing of B from A ?

A boat sails round a quadrangular course ABCD, starting from A to B in4km due east of A,C 3km due South of B and D 4km S50W from C.what is the distance and bearing of A from D?

You can take a look at my explanation below:

https://www.youtube.com/watch?v=vteDpAUthkM

Cheers,

Mr Will

I don’t know the Calculation at all and how can I coz is coming out in Waec(Exam)…pls I need your help sir or ma??????

Hi Joseph,

Just keep revising and make sure you understand the basic first.

You can take a look at your textbooks, study notes or even the summary notes and exercise examples I have written here.

Good luck for your exam and cheers!

Mr Will

how can some1 know the full understanding of bearing and distance calculation

1.A village is 10km on a bearing 0500 from a point o , how far is the village North of o .

2.The angle of elevation of the top of a building from a point 80m away on a level ground is 25°.calculate the height of the building to the nearest metre.

You can take a look at my explanation below:

https://youtu.be/A7f7mjSoeuY

Cheers,

Mr Will

The bearing of a town Y from X is 125°, find the bearing from X to Y

Nice and awesome thanks Mr. Will

You’re welcome 😃

Mr Will

three town A B C are such that the distance between A and B is 50km and the distance between A and B is 90km. if the bearing of B from A is 75degree and the bearing of C from A is 310degree find the @ distance between B and C (b)bearing of C and B

unable to solve this

B is 48 km West and 31 km south of A. Calculate the bearing from B to A.

Please help me with this questions.

An aeroplane X whose average speed is 50°km/hr leaves Kano air port at 7:00am and travels for 2 hours n a bearing 050°. It then changes its course and flies on a bearing 1200 to an airstrip A.

Another aeroplane Y leaves Kano airport at 10:00am and flies on a straight course to the airstrip A. Both planes arrives at the airstrip A at 11:30am. Calculate

(A)the average speed of Y to three significant figures

(B) the direction of flight Y to the nearest degree

please can you help me with this question P,Q,R,S are four locations on the same horizontal plane Q is on a bearing of 041 degrees from P and the distance is 40 kilometres S is 48 kilometres from R on a bearing of 074 degrees.R is directly due north of P and the distance between Q and R is 38 kilometres find

(a) the bearing of R from Q

(b) the distance between Q and S

(c) the distance between P and R

A ship leaves port p and sails on a bearing N50°E to a port Q 15km away. It then sails on a bearing of S45°E to port R, 20km away

pls u put only the question u did nt specify on what we are to look for

IS VERY GOOD

IS GOOD

Two ships A and B leaves a port simultaneously. A steams at 10kmh^-1 on the bearing of 215. After one hour, the bearing B from A is 260. Find the speed of B

If a ship leaves aport A and travels 215km on a bearing of 113 degrees to Port B and then travels on a bearing of 172 degrees to another Port C, 305km away.What will be the sketch of the ship’s journey?.Show the diagram sketch of the ship.

A car travels from town p for50km due north to town q . It then travels 80km on a bearing of 270 to town r. Calculate . The shortest distance from p to town r , the bearing of town r from p ,the bearing of p from r

A surveyor observes that the bearing of a beacon O from two other beacons P and Q are . N30degreeW and N70degreeE respectively. If the bearing of P from Q is 080degree and PQ = 300m. Calculate (a)the distance PQ (b)the distance of O from PQ (c)the bearing of Q from P

Why can’t you answer your question

A town b is due north of a,a third town c is 10km on a bearing of 020 from a .if b is ona bearing of 290 from c,calculate the distance bc and ab

A courtier bus travels from Head office in Lagos to a town B 210km away on a bearing 055° to deliver a mall.it then changes courses and moves to it’s branch office down in Lagos on a bearing of 220°. If the branch office is directly east of the head office.calculate correct to 3 significant figure The distance between the head office and the branch office