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Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Geometry and Measure

Bearings and Midpoints

Read, measure, draw and calculate three-figure bearings (including back bearings), and use coordinates to find the midpoint of a line segment.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum
  • Understand bearings as a way of giving a direction, and use them: read, measure, draw and calculate bearings. [8Gp.01]
  • Use coordinates to find the midpoint of a line segment. [8Gp.02]

Pilots, sailors and map readers need a precise way to say which direction to travel. 'North-east' is not exact enough — a bearing gives the direction as an angle. In the second part of the page you will use coordinates to find the exact middle of a line segment, its midpoint, without drawing anything.

1. Key words

Word / phraseMeaning / use
bearingالاتجاه بالزاوية (الاتجاه الملاحي) — an angle measured clockwise from North, written with 3 figures
North lineخط الشمال — a vertical line pointing North drawn at the starting point
clockwiseفي اتجاه عقارب الساعة — the direction the hands of a clock turn
three-figure bearingاتجاه بثلاثة أرقام — e.g. 045°, 120°, 300°
back bearingالاتجاه العكسي — the bearing for the return journey
scale drawingرسم بمقياس رسم — an accurate drawing with all lengths reduced in the same ratio
coordinatesالإحداثيات — (x, y): across first, then up or down
line segmentقطعة مستقيمة — the part of a line between two end points
midpointنقطة المنتصف — the point exactly halfway between two points

2. What is a bearing?

A bearing follows three rules:

  1. It is measured from North.
  2. It is measured clockwise.
  3. It is written with three figures (add zeros in front if needed): 5° is written 005°, 45° is written 045°.
N000°NE045°E090°SE135°S180°SW225°W270°NW315°clockwise
The eight compass points as three-figure bearings. East is 090° because it is a quarter turn clockwise from North.
DirectionNNEESESSWWNW
Bearing000°045°090°135°180°225°270°315°
قواعد الـ bearing
  • ابدأ دايماً من الشمال (N).
  • لف مع عقارب الساعة.
  • اكتب الزاوية بـ 3 أرقام: 45° تتكتب 045°.
  • «bearing of B from A» يعني ارسم خط الشمال عند A (النقطة اللي بعد كلمة from).

3. Measuring and drawing bearings

'The bearing of B from A' means: stand at A, face North, and turn clockwise until you face B. So the North line and the protractor go at the point after the word from.

  1. Draw (or find) the North line at the starting point.
  2. Join the starting point to the other point.
  3. Measure the angle clockwise from the North line to this line.
  4. If the angle is more than 180°, it is easier to measure the anticlockwise angle from North and subtract it from 360°.
  5. Write the answer with three figures.
N60°ABbearing of B from A = 060°N60°PQbearing of Q from P = 360° − 60° = 300°
Left: 60° clockwise from North gives 060°. Right: Q is 60° anticlockwise from North, so the clockwise angle is 360° − 60° = 300°.

Drawing a bearing: to draw a path on a bearing of 130° from P, draw a North line at P, put the protractor's centre on P with 0° on the North line, mark 130° going clockwise, and draw the line. For a bearing such as 250°, you can mark 180° (South) and then another 70° clockwise, or measure 360° − 250° = 110° anticlockwise from North.

Bearings and scale drawings: in a journey question, a bearing gives the direction and a scale gives the distance. Example: a boat sails 6 km on a bearing of 060°, using a scale of 1 cm to 1 km. Draw a North line, measure 60° clockwise, and mark a point 6 cm along the line.

4. Back bearings (the return journey)

If you know the bearing of B from A, you can work out the bearing of A from B without measuring. The two North lines are parallel, so the angle facts from the parallel-lines topic apply. The direction from B back to A is exactly opposite (a half turn, 180°) to the direction from A to B.

NN70°70°AB180° + 70° = 250°
The bearing of B from A is 070°. The two North lines are parallel, so the angle at B between South and BA is also 70° (alternate angles). The bearing of A from B is 180° + 70° = 250°.
  • If the bearing is less than 180°: back bearing = bearing + 180°. (070° → 250°)
  • If the bearing is more than 180°: back bearing = bearing − 180°. (300° → 120°)
  • Check: the two bearings always differ by exactly 180°.

5. The midpoint of a line segment

The midpoint is halfway along the line segment, so it is halfway across and halfway up. Halfway between two numbers is their mean (add and divide by 2). So:

Midpoint rule

The midpoint of (x₁, y₁) and (x₂, y₂) is ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 ). Add the x-coordinates and halve; add the y-coordinates and halve.

xy−4−3−2−1123456−11234560A(−3, 1)B(5, 5)M(1, 3)
A(−3, 1) and B(5, 5): x = (−3 + 5) ÷ 2 = 1 and y = (1 + 5) ÷ 2 = 3, so the midpoint is M(1, 3). The steps from A to M (4 across, 2 up) are the same as from M to B.

Working backwards: if you know one end A and the midpoint M, the other end B is the same step again from M. Step from A to M, then repeat it. As a rule: B = (2 × M's x − A's x, 2 × M's y − A's y).

نقطة المنتصف

اجمع الـ x واقسم على 2، واجمع الـ y واقسم على 2. خلي بالك من الإشارات السالبة: (−3 + 5) ÷ 2 = 1. ولو عندك طرف ونقطة المنتصف: كرّر نفس الخطوة من المنتصف علشان توصل للطرف التاني.

6. Worked examples

Worked example 1 — The angle measured clockwise from North at A to the line AB is 72°. Write the bearing of B from A. Then write the bearing of a point C that is due South-West of A. [2]

  1. Bearing of B from A = 072° (three figures).
  2. South-West is halfway between South (180°) and West (270°): 225°.

Worked example 2 — From a lighthouse L, a ship S is 50° anticlockwise from North. Work out the bearing of S from L. [1]

  1. Bearings are clockwise: 360° − 50° = 310°.

Common mistake: writing 050°. That would be 50° clockwise — on the other side of the North line.

Worked example 3 — (a) The bearing of B from A is 065°. Work out the bearing of A from B. (b) The bearing of D from C is 300°. Work out the bearing of C from D. [2]

  1. (a) 065° is less than 180°, so add 180°: 245°.
  2. (b) 300° is more than 180°, so subtract 180°: 120°.
  3. Check: 245 − 65 = 180 ✓ and 300 − 120 = 180 ✓

Worked example 4 — A boat leaves harbour H and sails 5 km on a bearing of 120°. Use a scale of 1 cm to 1 km to show its position. [2]

  1. Mark H and draw a North line through it.
  2. Put the protractor centre on H, 0° on the North line, and mark 120° clockwise (this is 30° past East).
  3. Draw a line from H through the mark and measure 5 cm along it (5 km at 1 cm to 1 km). Mark the boat B.
  4. Check: the bearing of H from B should be 120° + 180° = 300°.

Worked example 5 — Find the midpoint of A(−2, 7) and B(6, −1). [2]

  1. x: (−2 + 6) ÷ 2 = 4 ÷ 2 = 2.
  2. y: (7 + (−1)) ÷ 2 = 6 ÷ 2 = 3.
  3. Midpoint = (2, 3).

Common mistake: subtracting instead of adding, (6 − (−2)) ÷ 2 = 4. That is half the distance across, not the midpoint.

Worked example 6 — M(3, −1) is the midpoint of AB. A is the point (−1, 4). Find the coordinates of B. [2]

  1. From A to M: x goes from −1 to 3 (+4); y goes from 4 to −1 (−5).
  2. Do the same step again from M: x = 3 + 4 = 7; y = −1 − 5 = −6.
  3. B = (7, −6). Check: ((−1 + 7) ÷ 2, (4 + (−6)) ÷ 2) = (3, −1) ✓
Common mistakes
  • Measuring anticlockwise, or measuring from East or from the line AB instead of from North.
  • Putting the North line at the wrong point: 'bearing of B from A' — the North line goes at A.
  • Writing only two figures: 45° instead of 045°.
  • For back bearings, adding 180° to a bearing that is already more than 180° (giving more than 360°).
  • Midpoint: subtracting the coordinates, or forgetting to halve.
  • Sign errors with negative coordinates: −3 + 5 = 2, not −8.
How Checkpoint asks this
  • 'Measure the bearing of B from A.' — draw the North line if it is missing, measure clockwise, give three figures. Measure carefully to the nearest degree.
  • 'The bearing of P from Q is 070°. Work out the bearing of Q from P.' — no measuring needed: ±180°.
  • A map with a scale: find the real distance and the bearing, or plot a point from a bearing and a distance.
  • 'Work out the coordinates of the midpoint of AB.' — answer as (x, y) with brackets and a comma.
  • 'M is the midpoint of AB… find the coordinates of B.' — work backwards with the same step.
  • Explain/critique: spot a mistake in someone's bearing or midpoint method and correct it.

Practice

Remember
  • Bearings: from North, clockwise, three figures (045°, not 45°).
  • 'Bearing of B from A' — the North line and protractor go at A.
  • Back bearing: add 180° if the bearing is less than 180°, subtract 180° if it is more.
  • Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
  • To find an end point from the midpoint, repeat the step from the known end to the midpoint.
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