LEARNING OBJECTIVES
What you will be able to do
- use geometrical and circle vocabulary
- construct shapes and interpret nets
- work with scale drawings and bearings
- solve similarity problems
- identify line, rotational and solid symmetry
- justify angle calculations
- apply circle theorems
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Use precise language, construction, symmetry and angle reasoning to explain shapes and relationships.
Geometry is reasoning about space. A correct numerical answer is stronger when each step is supported by a named property.
Diagrams are not always drawn to scale, so use given measurements, markings and established theorems rather than appearance.
SECTION 01
Language, constructions and nets
A point has position but no size; a line extends in both directions; a line segment has two endpoints; and a plane is a flat surface extending in two dimensions. Parallel lines never meet, while perpendicular lines meet at 90°.
Triangles are classified by side lengths or angles. Special quadrilaterals are identified by combinations of parallel sides, equal sides, right angles and diagonal properties.
For a ruler-and-compass triangle construction, draw one side, then intersect arcs with radii equal to the other two sides. Leave the construction arcs visible. A net is a flat arrangement of faces that folds into a solid; shared edges must have equal lengths.
DETAILED EXPLANATION
- A face is flat, an edge is where faces meet and a vertex is a corner.
- A tangent touches a circle once; a chord joins two points on the circle.
- A sector lies between two radii and an arc; a segment lies between a chord and an arc.
| Shape | Defining properties |
|---|---|
| Square | four equal sides; four right angles |
| Rectangle | opposite sides equal and parallel; four right angles |
| Rhombus | four equal sides; opposite angles equal |
| Parallelogram | opposite sides equal and parallel; opposite angles equal |
| Kite | two pairs of equal adjacent sides |
| Trapezium | one pair of parallel sides |
Constructing a triangle with sides 7 cm, 5 cm and 4 cm
- Draw a 7 cm base with a ruler.
- From one endpoint draw an arc of radius 5 cm.
- From the other endpoint draw an arc of radius 4 cm.
- Join the intersection of the arcs to both endpoints.
Answer: The three fixed side lengths determine the triangle up to reflection.
SECTION 02
Scale drawings and bearings
A scale connects drawing length to real length. Convert to common units before calculating.
Bearings are measured clockwise from north and written with three digits from 000° to 360°.
Interpreting a map scale of 1 : 50 000
- A road measures 7.6 cm on the map.
- Actual distance = 7.6 × 50 000 = 380 000 cm.
- Convert: 380 000 cm = 3.8 km.
Answer: The road is 3.8 km long.
SECTION 03
Similarity and symmetry
Similar shapes have equal corresponding angles and proportional corresponding lengths. Congruent shapes have the same shape and size.
Triangles can be shown to be similar using equal corresponding angles, or by matching proportional sides with the included angle. Write vertices in corresponding order so the side ratios are unambiguous.
If the linear scale factor is k, area changes by k² and surface area also changes by k², while volume changes by k³.
Line symmetry reflects a figure onto itself; rotational order counts how many times it matches during one full turn. Prisms, cylinders, pyramids and cones may have planes or axes of symmetry.
Volumes of similar solids
- Two similar containers have heights in the ratio 3 : 5.
- The volume ratio is 3³ : 5³ = 27 : 125.
- The smaller volume is 216 cm³, so one ratio part is 216 ÷ 27 = 8 cm³.
- Larger volume = 125 × 8.
Answer: The larger volume is 1000 cm³.
SECTION 04
Angle facts and polygons
Angles around a point total 360°, angles on a straight line total 180°, vertically opposite angles are equal, and triangle angles total 180°.
For parallel lines, corresponding and alternate angles are equal; co-interior angles sum to 180°.
A polygon's exterior angles total 360°. Its interior sum can be split into triangles.
Finding the number of sides of a regular polygon
- Each interior angle is 165°.
- Each exterior angle is 180° − 165° = 15°.
- Number of sides = 360° ÷ 15°.
Answer: The polygon has 24 sides.
SECTION 05
Circle angle theorems
An angle in a semicircle is 90°. A radius is perpendicular to a tangent at the point of contact.
The angle at the centre is twice the angle at the circumference standing on the same arc. Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral sum to 180°. The angle between a tangent and chord equals the angle in the alternate segment.
DETAILED EXPLANATION
- A cyclic quadrilateral has all four vertices on the circle.
Using the centre theorem
- Chord AB subtends angle AOB = 124° at the centre.
- Angle ACB stands on the same chord AB at the circumference.
- The circumference angle is half the centre angle.
Answer: Angle ACB = 62°.
SECTION 06
Chord and tangent symmetry
Equal chords lie the same perpendicular distance from the centre. The perpendicular bisector of any chord passes through the centre.
Two tangents drawn from the same external point have equal lengths. Joining that point to the centre often creates congruent right-angled triangles.
Equal tangents from an external point
- PA and PB are tangents from P to a circle.
- PA = 3x + 4 and PB = 5x − 8.
- Set them equal: 3x + 4 = 5x − 8.
- 12 = 2x, so x = 6.
Answer: Each tangent has length 22 units.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Use exact geometrical vocabulary.
- Show construction arcs and use three-digit bearings.
- Similarity scales areas by k² and volumes by k³.
- Support angle answers with named reasons.
- Match circle theorems to the same arc, chord or tangent.
- Circle symmetry links centres, chords and equal tangents.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- use geometrical and circle vocabulary
- construct shapes and interpret nets
- work with scale drawings and bearings
- solve similarity problems
- identify line, rotational and solid symmetry
- justify angle calculations
- apply circle theorems