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4a · E4.1–E4.3

Geometrical terms, constructions and scale drawings

Topic 4 Geometry · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Use the vocabulary of points, lines, angles (acute, obtuse, reflex, interior, exterior), parallel and perpendicular lines, similar and congruent shapes.
  • Name and describe triangles, special quadrilaterals, polygons, solids and the parts of a circle.
  • Measure and draw lines and angles; construct a triangle from three sides with a ruler and compasses, showing the arcs.
  • Draw, use and interpret nets, including to find surface area and volume.
  • Draw and interpret scale drawings, and use three-figure bearings.

📚The mathematics

The vocabulary

  • Angles: acute (less than 90°), right (90°), obtuse (between 90° and 180°), reflex (between 180° and 360°).
  • Triangles: equilateral (3 equal sides), isosceles (2), scalene (none), right-angled.
  • Quadrilaterals: square, rectangle, rhombus (4 equal sides), parallelogram (2 pairs of parallel sides), trapezium (1 pair), kite (2 pairs of equal adjacent sides). Be ready to state properties of sides, angles, diagonals and symmetry.
  • Polygons: pentagon (5), hexagon (6), octagon (8), decagon (10); regular means all sides and angles equal.
  • Similar shapes are enlargements of each other; congruent shapes are identical in shape and size.
  • Solids: cube, cuboid, prism, cylinder, pyramid, cone, sphere, hemisphere, frustum; faces, edges and vertices.
A labelled circle showing its centre, a radius, a diameter, a chord, a tangent touching the circle at one point, a shaded sector between two radii, a shaded segment between a chord and the arc, and a minor arc.
The parts of a circle. A sector is bounded by two radii; a segment by a chord.

Constructing a triangle from three sides

  1. Draw the longest side with a ruler, and label its ends.
  2. Set the compasses to the second length and draw an arc from one end.
  3. Set them to the third length and draw an arc from the other end.
  4. Join the crossing point to both ends. Leave the arcs showing — they are the evidence the method was used.
Construction of triangle ABC with AB = 7 cm, AC = 5 cm and BC = 6 cm. AB is drawn horizontally; an arc of radius 5 cm is drawn from A and an arc of radius 6 cm from B, crossing at C, which is joined to A and B.
\( AB = 7 \), \( AC = 5 \), \( BC = 6 \) (cm): the two arcs meet at \(C\). Constructing perpendicular and angle bisectors is not required by the current syllabus.

Scale drawings and bearings

A scale such as \( 1 : 50\,000 \) means 1 cm on the map is 50 000 cm = 500 m in real life. Convert units at the end: 100 cm = 1 m, 1000 m = 1 km.

A bearing is an angle measured clockwise from north, written with three figures (\( 065^{\circ} \), not \( 65^{\circ} \)). “The bearing of \(B\) from \(A\)” means stand at \(A\), face north, turn clockwise to face \(B\). The back bearing (of \(A\) from \(B\)) differs by \( 180^{\circ} \).

Two points A and B with a north line drawn at each. The bearing of B from A is 65 degrees, measured clockwise from north at A. At B, the bearing of A from B is 245 degrees, measured clockwise from north at B; the two north lines are parallel, so 65 plus 180 gives 245.
The north lines are parallel, so the angles are alternate: bearing of \(A\) from \(B\) \( = 065^{\circ} + 180^{\circ} = 245^{\circ} \).

✏️Worked example (non-calculator)

(a) The bearing of \(B\) from \(A\) is \( 065^{\circ} \). Find the bearing of \(A\) from \(B\). [1] (b) On a map with scale \( 1 : 50\,000 \), two villages are 7.4 cm apart. Find the real distance in kilometres. [2] (c) A cuboid measures 5 cm by 3 cm by 2 cm. Find the area of its net. [2]

(a) \( 65 + 180 = 245 \): bearing \( 245^{\circ} \).

(b) \( 7.4 \times 50\,000 = 370\,000 \) cm \( = 3700 \) m \( = 3.7 \) km.

(c) The net is the surface: \( 2(5 \times 3 + 5 \times 2 + 3 \times 2) = 2(15 + 10 + 6) = 62 \) cm².

Check it. Every bearing and its back bearing differ by exactly \( 180^{\circ} \), and both lie between \( 000^{\circ} \) and \( 360^{\circ} \): 065 and 245 ✓. If adding 180 goes past 360, subtract 180 instead.
Measuring from the wrong point. “From \(A\)” means the north line is drawn at \(A\). Measuring at \(B\) gives the back bearing, a different answer.

📝Practise

Questions in the style of the current papers.

1. (Non-calculator.) A quadrilateral has four equal sides, and its diagonals cross at right angles, but its angles are not all 90°. What is it? Write down one more property. [2]
A rhombus. Also: opposite sides parallel; opposite angles equal; diagonals bisect each other and the angles; 2 lines of symmetry; rotational symmetry of order 2.
2. (Non-calculator.) The bearing of \(Q\) from \(P\) is \( 310^{\circ} \). Find the bearing of \(P\) from \(Q\). [1]
\( 310 - 180 = 130 \): \( 130^{\circ} \). (Adding 180 would pass 360.)
3. (Non-calculator.) On a map with scale \( 1 : 25\,000 \), how many centimetres represent 2.5 km? [2]
2.5 km \( = 250\,000 \) cm; \( 250\,000 \div 25\,000 = 10 \) cm.
4. (Non-calculator.) Write down the number of faces, edges and vertices of a triangular prism. [2]
5 faces (2 triangles, 3 rectangles), 9 edges, 6 vertices.
5. (Non-calculator.) Describe the steps to construct triangle \(PQR\) with \( PQ = 8 \) cm, \( PR = 6 \) cm and \( QR = 5 \) cm. [3]
Draw \( PQ = 8 \) cm. With compasses at \(P\), radius 6 cm, draw an arc; with compasses at \(Q\), radius 5 cm, draw an arc crossing the first. The crossing point is \(R\); join \(PR\) and \(QR\), leaving the arcs visible.
6. (Calculator.) A ship sails 12 km due east, then 5 km due south. Find its distance from the start, and the bearing of its final position from the start. [4]
Distance \( \sqrt{12^{2} + 5^{2}} = 13 \) km. The angle south of east is \( \tan^{-1}\tfrac{5}{12} = 22.6^{\circ} \), so the bearing is \( 090^{\circ} + 22.6^{\circ} = 112.6^{\circ} \). (A sketch with a north line at the start makes this clear — see topic 6a.)

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • GeoGebra — practise ruler-and-compass constructions interactively
  • Corbettmaths — bearings and scale drawings