Hard GCSE Bearings and scale drawings Questions

Recall what a three-figure bearing is.

A bearing is the angle turned CLOCKWISE from the north line, written with three figures — so an angle of $8{}^\circ$ becomes $008^\circ$.

Note that both bearings are measured at A from the same north line.

Both directions are measured from the north line at $A$, so angle $BAC$ is simply the difference between the two bearings.

Subtract the smaller bearing from the larger one.

The two directions are $225^\circ$ apart one way round the north line at $A$.

Take the non-reflex angle.

The full turn at $A$ splits into $225^\circ$ and $135^\circ$. Angle $BAC$ is the one that is not reflex, so it is $135^\circ$.

Check what the answer is measured in.

The answer is the SIZE of an angle, so it is written $135^\circ$ and NOT as a three-figure bearing. Only directions get three figures.

Check the size of the angle is sensible.

An angle inside a triangle is always between $0^\circ$ and $180^\circ$, and $135^\circ$ is.

Check the angle asked for is the one that has been found.

Two angles meet at that point: $135^\circ$ and the reflex angle $225^\circ$. The question asks for the angle of the triangle, which is the smaller one, $135^\circ$.

Check the angle and its reflex angle add to a full turn.

The angle $135^\circ$ and the reflex angle around the same point add up to a full turn of $360^\circ$, which they do.

Do not write this answer with three figures.

Three-figure notation is for BEARINGS, which are directions. This answer is the size of an angle, so it is just $135^\circ$.

Check that the two north lines are parallel.

North is the same direction everywhere on the page, so the north line at one point is parallel to the north line at the other. That is what lets you use the parallel-line angle facts.

Check the angle was measured from north and not from east.

A bearing always starts at the north line and turns clockwise. Starting anywhere else, or turning the other way, gives the wrong bearing.

Check the angle on an accurate drawing.

On a drawing made to scale, measuring the angle with a protractor should give $135^\circ$. That is a useful check but not a proof — the working is the proof.

Note that a bearing does not depend on how far apart the points are.

A bearing records a DIRECTION. Moving the second point twice as far away along the same line does not change the bearing at all.

Check the answer against a rough sketch.

Drawing the north line and the direction roughly to scale is the quickest way to catch an answer that points the wrong way. The sketch does not need to be accurate to do that job.

State the correct option.

Angle $BAC = 135^\circ$.

Say what the scale means.

A scale of $1 : 50000$ means every length in real life is $50000$ times the matching length on the map, with both measured in the same unit.

Turn the map distance into a real distance in centimetres.

The real distance is $50000$ times the map distance, so multiply: $6 \times 50000$ cm.

Recall how centimetres and kilometres are linked.

There are $100$ cm in a metre and $1000$ m in a kilometre, so there are $100 \times 1000 = 100\,000$ cm in a kilometre.

Turn the real distance in centimetres into kilometres.

Dividing by $100000$ converts centimetres into kilometres, so the whole calculation is $6 \times 50000 \div 100000$.

Check the value the correct calculation gives.

The correct calculation gives $3$ km, which is a sensible distance between two towns. The other options give values that are far too big or far too small.

Check the units of the answer.

The question asks for the real distance in km, so the answer is $3\text{ km}$.

Recall how centimetres and kilometres are linked.

There are $100$ cm in a metre and $1000$ m in a kilometre, so there are $100 \times 1000 = 100\,000$ cm in a kilometre.

Check the two lengths were in the same unit before the scale was used.

A scale such as $1 : 25\,000$ only compares like with like. Both lengths must be turned into the same unit before the multiplication, and only then is the answer converted to the unit asked for.

Check the size of the answer is sensible.

The real distance must be a positive length, and $3$ km is a realistic size for it.

Check the answer by working backwards.

Reversing every step — dividing where you multiplied, and multiplying where you divided — must take $3$ km back to the length the question started from.

Note the commonest mistake with scales.

Multiplying when you should divide turns a real distance into an impossible one. Ask first whether the answer should be bigger or smaller than the length you were given.

Check the conversion factor was applied only once.

It is easy to divide by $100$ and then by $1000$ and then by $100\,000$ again. Each conversion between units happens exactly once.

Check the answer is written to a sensible accuracy.

The numbers in the question are exact, so $3$ km is exact too and needs no rounding.

State the rule this question uses.

Every map-scale question is this one rule, used forwards to get a real distance and backwards to get a map distance.

State the correct calculation.

The real distance is $6 \times 50000 \div 100000 = 3$ km.

Recall what a three-figure bearing is.

A bearing is the angle turned CLOCKWISE from the north line, written with three figures — so an angle of $8{}^\circ$ becomes $008^\circ$.

Note that the statement asked about is the back bearing.

The bearing given is measured at $P$. The statements are all about the bearing measured at $Q$, looking back at $P$.

State the rule for a back bearing.

The north lines at the two points are parallel, so the two bearings differ by exactly $180^\circ$: add $180^\circ$ if the bearing is less than $180^\circ$, and subtract $180^\circ$ if it is more.

Apply the back bearing rule.

$283 - 180 = 103$, so the bearing of $P$ from $Q$ is $103^\circ$.

Reject the statements that give any other bearing.

Only one statement gives $103^\circ$; every other statement names a different direction and so is false.

Write the answer as a three-figure bearing.

Bearings are always written with three figures, putting in zeros at the front if they are needed. So the answer is written $103^\circ$.

Check the answer is a possible bearing.

A bearing is measured clockwise from north and comes all the way back to north after a full turn, so every bearing lies between $000^\circ$ and $360^\circ$. $103^\circ$ does.

Check which part of the compass the direction points into.

$103^\circ$ lies between $090^\circ$ (east) and $180^\circ$ (south), so the direction points somewhere between east and south. A sketch should agree with that.

Check the answer by reversing it.

Turning through a further $180^\circ$ from $103^\circ$ leads back to $283^\circ$, which is where the reverse journey points. The two bearings differ by $180^\circ$, as they must.

Note the commonest mistake in bearings questions.

Bearings are measured CLOCKWISE from north. Measuring anticlockwise instead would give $257^\circ$, which is a different direction entirely.

Check the units of the answer.

A bearing is an angle, so it is measured in degrees. The answer is $103^\circ$, not $103$ of anything else.

Compare the answer with the nearest compass point.

The nearest compass point to $103^\circ$ is east ($090^\circ$), so the direction should look roughly east on a sketch.

Compare the bearing with the four main compass directions.

North is $000^\circ$, east is $090^\circ$, south is $180^\circ$ and west is $270^\circ$. Placing $103^\circ$ among these is a fast check that the direction is roughly right.

Check that the two north lines are parallel.

North is the same direction everywhere on the page, so the north line at one point is parallel to the north line at the other. That is what lets you use the parallel-line angle facts.

State the correct statement.

The bearing of $P$ from $Q$ is $103^\circ$.

Write down the bearing that is given, and note that the north lines at A and B are parallel.

Standing at $A$ and facing north, you turn $172^\circ$ clockwise to look at $B$. North points the same way at $B$, so the two north lines are parallel and the two bearings must differ by $180^\circ$.

Add or subtract 180 degrees to reverse the bearing.

$172^\circ$ is less than $180^\circ$, so add $180^\circ$: $172 + 180 = 352$.

Recall what a three-figure bearing is.

A bearing is the angle turned CLOCKWISE from the north line, written with three figures — so an angle of $8{}^\circ$ becomes $008^\circ$.

State the rule for a back bearing.

The north lines at the two points are parallel, so the two bearings differ by exactly $180^\circ$: add $180^\circ$ if the bearing is less than $180^\circ$, and subtract $180^\circ$ if it is more.

Check the choice of adding or subtracting was the right one.

Adding $180^\circ$ to a bearing above $180^\circ$ would push the answer past a full turn, and subtracting from a bearing below $180^\circ$ would push it below zero. Here $172^\circ$ is below $180^\circ$, so adding is right.

Write the answer as a three-figure bearing.

Bearings are always written with three figures, putting in zeros at the front if they are needed. So the answer is written $352^\circ$.

Check the answer is a possible bearing.

A bearing is measured clockwise from north and comes all the way back to north after a full turn, so every bearing lies between $000^\circ$ and $360^\circ$. $352^\circ$ does.

Check which part of the compass the direction points into.

$352^\circ$ lies between $270^\circ$ (west) and $000^\circ$ (north), so the direction points somewhere between west and north. A sketch should agree with that.

Check the answer by reversing it.

Turning through a further $180^\circ$ from $352^\circ$ leads back to $172^\circ$, which is where the reverse journey points. The two bearings differ by $180^\circ$, as they must.

Note the commonest mistake in bearings questions.

Bearings are measured CLOCKWISE from north. Measuring anticlockwise instead would give $008^\circ$, which is a different direction entirely.

Check the units of the answer.

A bearing is an angle, so it is measured in degrees. The answer is $352^\circ$, not $352$ of anything else.

Compare the answer with the nearest compass point.

The nearest compass point to $352^\circ$ is north ($000^\circ$), so the direction should look roughly north on a sketch.

Compare the bearing with the four main compass directions.

North is $000^\circ$, east is $090^\circ$, south is $180^\circ$ and west is $270^\circ$. Placing $352^\circ$ among these is a fast check that the direction is roughly right.

Check that the two north lines are parallel.

North is the same direction everywhere on the page, so the north line at one point is parallel to the north line at the other. That is what lets you use the parallel-line angle facts.

State the bearing of A from B.

The bearing of $A$ from $B$ is $352^\circ$.

Mark both bearings from the same north line at A, and subtract them.

Both bearings are measured at $A$ from the same north line, so the two directions are $155^\circ$ apart going one way round that north line.

Take the angle that is not reflex.

The two directions cut the full turn at $A$ into $155^\circ$ and $205^\circ$. Angle $BAC$ is the angle of the triangle, so it is the one that is not reflex: $155^\circ$.

Recall what a three-figure bearing is.

A bearing is the angle turned CLOCKWISE from the north line, written with three figures — so an angle of $8{}^\circ$ becomes $008^\circ$.

Say why the difference of the bearings is the angle.

Both bearings start from the SAME north line at $A$, so subtracting one from the other removes the north line entirely and leaves the angle between $AB$ and $AC$.

Check the subtraction.

Adding the difference back onto the smaller bearing returns the larger one, so $155^\circ$ is right.

Check what the answer is measured in.

The answer is the SIZE of an angle, so it is written $155^\circ$ and NOT as a three-figure bearing. Only directions get three figures.

Check the size of the angle is sensible.

An angle inside a triangle is always between $0^\circ$ and $180^\circ$, and $155^\circ$ is.

Check the angle asked for is the one that has been found.

Two angles meet at that point: $155^\circ$ and the reflex angle $205^\circ$. The question asks for the angle of the triangle, which is the smaller one, $155^\circ$.

Check the angle and its reflex angle add to a full turn.

The angle $155^\circ$ and the reflex angle around the same point add up to a full turn of $360^\circ$, which they do.

Do not write this answer with three figures.

Three-figure notation is for BEARINGS, which are directions. This answer is the size of an angle, so it is just $155^\circ$.

Check that the two north lines are parallel.

North is the same direction everywhere on the page, so the north line at one point is parallel to the north line at the other. That is what lets you use the parallel-line angle facts.

Check the angle was measured from north and not from east.

A bearing always starts at the north line and turns clockwise. Starting anywhere else, or turning the other way, gives the wrong bearing.

Check the angle on an accurate drawing.

On a drawing made to scale, measuring the angle with a protractor should give $155^\circ$. That is a useful check but not a proof — the working is the proof.

Note that a bearing does not depend on how far apart the points are.

A bearing records a DIRECTION. Moving the second point twice as far away along the same line does not change the bearing at all.

State the size of angle BAC.

Angle $BAC = 155^\circ$.