Geometrical Terms, Constructions and Scale Drawings at Extended Level
This unit is the toolkit for the whole Geometry strand. You need the right words, a steady pair of compasses, and confidence with scales and bearings.
Key words (E4.1). Point, vertex, line, plane, parallel, perpendicular. Angles: acute (less than 90°), right (90°), obtuse (between 90° and 180°), reflex (between 180° and 360°). Interior and exterior angles. A bearing is an angle measured clockwise from north.
Congruent or similar? Congruent shapes are exactly the same shape AND size (all matching sides and angles equal). Similar shapes are the same shape but can be a different size: matching angles are equal and matching sides are all in the same ratio, the scale factor. Triangles with sides 3, 4, 5 cm and 6, 8, 10 cm are similar with scale factor 2.
Triangles: equilateral (3 equal sides, all angles 60°), isosceles (2 equal sides, 2 equal base angles), scalene (no equal sides), right-angled (one 90° angle).
Quadrilaterals — know the diagonals too:
- Square: 4 equal sides, 4 right angles; diagonals equal, bisect each other at 90°.
- Rectangle: opposite sides equal, 4 right angles; diagonals equal and bisect each other (not at 90°).
- Rhombus: 4 equal sides, opposite angles equal; diagonals bisect each other at 90° but are not equal.
- Parallelogram: opposite sides equal and parallel, opposite angles equal; diagonals bisect each other.
- Kite: two pairs of equal adjacent sides; one pair of equal opposite angles; diagonals cross at 90°, only one is bisected.
- Trapezium: exactly one pair of parallel sides.
Polygons: pentagon (5), hexagon (6), octagon (8), decagon (10). A regular polygon has all sides equal AND all angles equal.
Solids and nets: cube, cuboid, prism, cylinder, pyramid, cone, sphere, hemisphere, frustum. Know faces, edges and vertices. A net is the flat shape that folds up into the solid. Quick check for any solid with flat faces: faces + vertices − edges = 2.
Circle words: centre, radius, diameter, circumference, semicircle, chord, tangent, arc, sector, segment.
Constructions (E4.2). You must measure and draw lines and angles accurately, and construct a triangle from its three sides using a ruler and a pair of compasses only. Leave all construction arcs showing — they earn the marks.
What the syllabus asks for: construct a triangle from its three sides using a ruler and compasses only (arcs showing), build shapes such as a rhombus from two triangles, and draw and use nets. Constructions of perpendicular bisectors and angle bisectors are not required in IGCSE 0580.
Scale drawings and bearings (E4.3). A map scale 1 : n means 1 cm on the map is n cm in real life. Convert with care: 100 cm = 1 m and 100 000 cm = 1 km. For areas, multiply by n² (because both length and width are scaled). Bearings are always written with three figures, from 000° to 360°.
Triangle P has sides 3 cm, 4 cm, 5 cm. Triangle Q has sides 6 cm, 8 cm, 10 cm. Triangle R has sides 5 cm, 3 cm, 4 cm. Which triangles are congruent and which are similar?
P and R have the same three lengths, just listed in a different order. So P and R are congruent.
Match smallest with smallest: \(\dfrac{6}{3} = \dfrac{8}{4} = \dfrac{10}{5} = 2\). Every ratio is the same, so Q is similar to P (and to R) with scale factor 2. Q is not congruent to P, because its size is different.
Using a ruler and compasses only, construct triangle ABC with AB = 7 cm, AC = 5 cm and BC = 6 cm. Then measure angle BAC.
1. Draw AB = 7 cm with a ruler. 2. Open the compasses to 5 cm, put the point on A and draw an arc above AB. 3. Open to 6 cm, put the point on B and draw an arc that crosses the first one. 4. The crossing point is C. Join AC and BC. Do not rub out the arcs.
Measured with a protractor, angle BAC ≈ 57° (an answer within 1° or 2° is accepted).
Check: the largest angle is always opposite the longest side. AB = 7 cm is longest, so angle ACB is the largest angle.
A map of the Gaborone area has scale 1 : 25 000. (a) A road is 8.4 cm long on the map. Find the real length in km. (b) A field has area 6 cm² on the map. Find its real area in km².
(a) 8.4 × 25 000 = 210 000 cm. Divide by 100 000 to get km: 2.1 km.
(b) Area factor = 25 000² = 625 000 000. So 6 × 625 000 000 = 3 750 000 000 cm².
\(1\text{ km}^2 = 100\,000^2\text{ cm}^2 = 10^{10}\text{ cm}^2\), so \(\dfrac{3\,750\,000\,000}{10^{10}} = 0.375\text{ km}^2\)
Real area = 0.375 km². Squaring the scale is the step most students miss.
The bearing of Francistown from Nata is 136°. Find the bearing of Nata from Francistown.
136° is less than 180°, so add 180°: 136° + 180° = 316°.
Picture it: Francistown is roughly south-east of Nata, so Nata must be roughly north-west of Francistown. 316° is in the north-west, so the answer makes sense.
A plane flies from S for 8 km on a bearing of 060° to P. It then flies 6 km on a bearing of 150° to F. Make a scale drawing (1 cm to 1 km) and find the distance and bearing of F from S.
Draw a north line at S, measure 60° clockwise and draw SP = 8 cm. Draw a new north line at P, measure 150° clockwise and draw PF = 6 cm. Join SF.
The two directions differ by 150° − 60° = 90°, so angle SPF is a right angle. Measuring (or checking with Pythagoras): SF = 10 km. Measure the bearing of F clockwise from the north line at S: about 097°.
Q1. Triangle A has sides 4 cm, 6 cm and 7 cm. Triangle B has sides 12 cm, 18 cm and 21 cm. Triangle C has sides 7 cm, 4 cm and 6 cm.
(a) Which two triangles are congruent? (b) Which triangle is similar but not congruent to A? Give the scale factor.
👀 Show answer
(b) 124 = 186 = 217 = 3 ✓ M1
B, scale factor 3 ✓ A1
Q2. Name the quadrilateral described.
(a) Its diagonals are equal in length and bisect each other at 90°.
(b) All four sides are equal, but its diagonals are different lengths.
(c) It has exactly one pair of parallel sides.
👀 Show answer
(b) rhombus ✓ B1
(c) trapezium ✓ B1
Q3. (a) A net is made of 2 congruent triangles and 3 rectangles. Name the solid it folds into.
(b) Write down the number of faces, edges and vertices of this solid.
👀 Show answer
(b) faces 5, edges 9, vertices 6 ✓ B2
Q4. Using a ruler and compasses only, construct triangle PQR with PQ = 8 cm, PR = 6 cm and QR = 5 cm. Leave in your construction arcs. Then measure angle PRQ.
👀 Show answer
arcs cross at R, triangle completed, arcs visible ✓ B1
angle PRQ = 93° (accept 91° to 95°) ✓ B1
Q5. A map has scale 1 : 50 000. The road from Mogoditshane to a village is 7.3 cm on the map. Find the real distance in kilometres.
👀 Show answer
365 000 ÷ 100 000 = 3.65 km ✓ A1
Q6. A map has scale 1 : 20 000. A dam on the map has area 12.5 cm². Find the real area of the dam in km².
👀 Show answer
12.5 × 20 0002 = 5 000 000 000 cm2 ✓ M1
÷ 1010 → 0.5 km2 ✓ A1
Q7. (a) The bearing of Maun from Nata is 285°. Find the bearing of Nata from Maun.
(b) The bearing of Q from P is 042°. Find the bearing of P from Q.
👀 Show answer
(b) 042 + 180 = 222° ✓ B1
Q8. B is 12 km from A on a bearing of 070°. C is 5 km from B on a bearing of 160°.
(a) Find the bearing of A from B. (b) Show that angle ABC = 90°. (c) Find the distance AC. (d) Find the bearing of C from A.
👀 Show answer
(b) angle ABC = 250 − 160 = 90° ✓ B1
(c) AC = √(122 + 52) = √169 = 13 km ✓ B1
(d) angle BAC: tan = 512 → 22.6° ✓ M1
070 + 22.6 = 092.6° (accept 093°) ✓ A1
Q9. On a scale drawing with scale 1 : 200, a classroom measures 4.6 cm by 3.5 cm. Find (a) the real length and width in metres, (b) the real floor area in m².
👀 Show answer
(b) 9.2 × 7 = 64.4 m2 ✓ B1
- Mixing up congruent and similar: congruent = same shape AND size; similar = same shape, sides in one fixed ratio.
- Rubbing out construction arcs: the arcs are your evidence. No arcs, no construction marks.
- Arcs centred on the wrong point: PR = 6 cm means an arc of radius 6 cm centred on P, not on Q.
- Scaling an area by n instead of n²: with a 1 : 20 000 map, areas scale by 20 000².
- Two-figure bearings or anticlockwise angles: bearings go clockwise from north and always have three figures (042°, not 42°).
- Wrong unit change: 1 km = 100 000 cm and 1 km² = 1010 cm².
- Paper 2 (no calculator): 1–2 mark items — name a shape from its properties, a back bearing, a map distance.
- Paper 4 (calculator): “Construct…”, “Using a ruler and compasses only…” (2–3 marks) and multi-part bearings questions that run on into Pythagoras and trigonometry (4–6 marks).
- Command words: “Construct” means use compasses and leave arcs; “Measure” means read from your own accurate drawing (an answer within 1–2° or 1–2 mm is accepted).
- Map scale area questions (1 : n with n²) appear often as the final part of a scale question — a typical 3-mark trap.