Ch 1 · 4.1–4.3 Geometrical Terms, Constructions, Scale Drawings and Bearings · O Level Mathematics 4024
Topic 1 of 4 in Geometry — O Level Mathematics 4024 — 58 lessons.
Introduction
Architects draw plans to scale, sailors steer by bearings, and packaging designers fold flat nets into boxes. This lesson sets out the vocabulary of geometry and the practical drawing skills: accurate construction with ruler and compasses, scale drawings and bearings. (Syllabus sections 4.1, 4.2 and 4.3.)
Key ideas — Vocabulary
🧑🏫 Teacher's explanation
Geometry is the language of shape and space. Instead of just seeing a 'box', we see a 'cuboid' with specific 'vertices' and 'edges'. Instead of seeing 'the same thing', we distinguish between shapes that are 'congruent' (identical twins) and shapes that are 'similar' (a photo and its enlargement).
📖 Definitions
🔍 Worked example — Easy
A triangle has side lengths 5 ext cm, 5 ext cm, and 8 ext cm. Classify the triangle by its sides and determine if it is isosceles.
🔍 Worked example — Standard
Point A is at a bearing of 040^° from Point B. Calculate the bearing of B from A.
🔍 Worked example — Exam-style
A photo of a rectangle has a length of 10 ext cm and a width of 6 ext cm. The actual rectangle has a length of 50 ext m. Calculate the scale factor used for the drawing.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Constructing a triangle from three sides
To draw a triangle with sides 7 cm, 6 cm and 5 cm: rule the 7 cm side; set the compasses to 6 cm and draw an arc from one end; set them to 5 cm and draw an arc from the other end; join the intersection to both ends. Leave the construction arcs visible. A ruler is used for every straight edge. (Perpendicular and angle bisector constructions are not required.)
🧑🏫 Teacher's explanation
Imagine you are building a tripod for a camera. If you know exactly how long each leg needs to be, there is only one specific way those legs can meet to form a stable base. In geometry, if you are given three specific side lengths, you are essentially 'locking' the shape into one possible configuration.
📖 Definitions
🔍 Worked example — Easy
Construct a triangle with side lengths 5 cm, 5 cm, and 8 cm. Describe the steps.
🔍 Worked example — Standard
Construct a scalene triangle with sides 7 cm, 6 cm, and 5 cm. State the type of triangle formed.
🔍 Worked example — Exam-style
A student is asked to construct a triangle with sides 4 cm, 3 cm, and 2 cm. They draw a 4 cm base, then arcs of 3 cm and 2 cm. The arcs do not meet. Explain why.
⚠️ Common errors
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🎯 Exam question
Key ideas — Nets
A net is a flat shape that folds into a solid. A cube has 11 different nets of six squares. Use measurements from a net to find a surface area (add the areas of the faces) or a volume.
🧑🏫 Teacher's explanation
Imagine you have a cardboard box that you want to flatten out to store it in a drawer. That flat, unfolded shape is what we call a 'net'. When you fold the edges of the net back up, the flat faces meet to form a 3D solid like a cube or a prism.
📖 Definitions
🔍 Worked example — Easy
A net of a cube consists of six squares. Each square has a side length of 4 cm. Calculate the total surface area of the cube.
🔍 Worked example — Standard
A net of a triangular prism is shown. The prism has two identical triangular faces with a base of 6 cm and a height of 4 cm. The three rectangular faces have dimensions 6\ ext cm × 5 cm, 5 cm × 5 cm, and 5 cm × 5 cm. Calculate the total surface area.
🔍 Worked example — Exam-style
A net of a square-based pyramid is shown. The base is a square with side length 10 cm. The four triangular faces are isosceles triangles with a base of 10 cm and a slant height (the height of the triangle) of 12 cm. Calculate the total surface area of the pyramid.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Scale drawings
A scale such as 1 : 200 means 1 cm on the drawing represents 200 cm = 2 m in real life. Multiply to go from the drawing to real life; divide to go back. Angles are not changed by scaling.
🧑🏫 Teacher's explanation
Imagine you are looking at a massive skyscraper. You cannot draw it on a piece of paper at its actual size, or the paper would need to be kilometers long! Instead, you use a scale drawing. A scale is a ratio that tells you how much you have 'shrunk' the real world to fit on your page.
📖 Definitions
🔍 Worked example — Easy
A map has a scale of 1:50,000. If the distance between two towns on the map is 4 cm, what is the actual distance in kilometers?
🔍 Worked example — Standard
A model car is built to a scale of 1:24. The actual length of the car is 4.8 m. Calculate the length of the model car in centimeters.
🔍 Worked example — Exam-style
A rectangular field is represented on a plan with a scale of 1:500. On the plan, the field measures 12 cm by 8 cm. Calculate the actual area of the field in square meters (m^2).
⚠️ Common errors
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🎯 Exam question
Key ideas — Three-figure bearings
A bearing is measured clockwise from north, written with three figures: 045^°, 250^°. "The bearing of B from A" means stand at A, face north, turn clockwise to B.
🧑🏫 Teacher's explanation
Imagine you are standing on a giant compass rose on the ground. To give someone directions, you don't just say 'turn left'; you need a standard starting point. In navigation, we always start facing North (000^°) and turn clockwise.
📖 Definitions
🔍 Worked example — Easy
A ship sails from Port A to Port B on a bearing of 050^\°. Calculate the bearing of Port A from Port B.
🔍 Worked example — Standard
The bearing of a lighthouse from a ship is 310^\°. Find the bearing of the ship from the lighthouse.
🔍 Worked example — Exam-style
Point X is due South of Point Y. Point Z is due East of Point X. Find the bearing of Z from Y.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples
On a plan with scale 1 : 50, a room is 9.6 cm long. Real length = 9.6 × 50 = 480 cm = 4.8 m.
🔍 Worked example — Easy
A map has a scale of 1 : 25,000. On the map, the distance between two towns is 4 cm. Calculate the actual distance in kilometers.
🔍 Worked example — Standard
A ship sails from port A on a bearing of 070^° to reach port B. Find the bearing of A from B.
🔍 Worked example — Exam-style
On a plan with scale 1 : 200, a rectangular garden is 5 cm wide and 8 cm long. Calculate the actual area of the garden in m^2.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) a trapezium. (2) 310^° - 180^° = 130^°. (3) 320 000 ÷ 25 000 = 12.8 cm. (4) rule 8 cm; arcs of radius 6 cm from each end; join the intersection to both ends (an isosceles triangle), leaving the arcs.