IGCSE Mathematics: Geometry — Revision Notes

Condensed for the final weeks. Pair these notes with the Geometry practice questions for worked exam-style application.

Tier note: items marked (Extended) are Extended only in the 0580 syllabus: area and volume scale factors of similar shapes (E4.4), angle properties of irregular polygons (E4.6; Core covers regular polygons), the four circle theorems beyond the Core pair (E4.7: angle at the centre, angles in the same segment, cyclic quadrilaterals, alternate segment) and the symmetry properties of chords and tangents (E4.8). The Core circle theorems are the angle in a semicircle and the angle between a tangent and a radius (C4.7). Everything else is Core. Constructing a perpendicular bisector or an angle bisector is not required at either tier (C4.2 and E4.2 both say so); those two methods are kept below as background only. Extended also adds the symmetry of 3D shapes (E4.5) and showing that two triangles are similar (E4.4); these are not covered on this page.

Geometrical terms and constructions

Bisector – a line dividing an angle or a line segment into two equal parts. Congruent shapes are identical in shape and size; similar shapes are identical in shape only, one a scaled copy of the other.

Constructing a triangle from its three sides (ruler and compasses only): draw the longest side with a ruler; set the compasses to the second length and draw an arc from one end of that side; set them to the third length and draw an arc from the other end; join the point where the two arcs cross to both ends. Leave the construction arcs visible: the syllabus says they must be shown, so do not rub them out.

Background only (the compass constructions are not required for 0580; the perpendicular bisector itself still appears as an Extended term and in E4.8):

Constructing a perpendicular bisector of AB: open compasses to more than half of AB; with the point on A, draw arcs above and below the line; repeat from B with the same radius so the arcs cross; join the two crossing points with a straight line.

Constructing the bisector of an angle: with the compass point on the angle’s vertex, draw an arc crossing both arms; from each of those two crossing points, draw a further arc of equal radius so the two new arcs intersect; join the vertex to that intersection point – this line bisects the angle exactly.

Scale drawings

A map scale such as 1 : 25 000 means one unit on the map represents 25 000 of the same unit in reality.

Map distance 4.4 cm, scale 1 : 25 000
real distance = 4.4 x 25 000 = 110 000 cm = 1100 m = 1.1 km

Always convert the final answer into the units the question asks for.

Similarity

Similar shapes have equal angles and sides in the same ratio, so a missing length is found by multiplying the corresponding length by the scale factor.

(Extended) If the linear scale factor between two similar shapes is k, then:

area scale factor   = k^2
volume scale factor = k^3

AB = 2 cm corresponds to PQ = 5 cm  ->  linear scale factor = 5/2 = 2.5
Area of ABC = 12 cm^2
Area of PQR = 12 x 2.5^2 = 12 x 6.25 = 75 cm^2

(Extended) Finding a linear scale factor from a volume ratio needs a cube root; from an area ratio, a square root.

Symmetry

Rotational symmetry (order) counts how many times a shape maps onto itself in one full 360-degree turn. Line symmetry counts the number of mirror lines. The two are not always equal – a non-rectangular parallelogram has rotational symmetry order 2 (it maps onto itself after a 180-degree turn) but no lines of symmetry at all, while a regular hexagon has both rotational symmetry order 6 and 6 lines of symmetry (equal, in that case, but not in general). A (non-rhombus) kite has exactly one line of symmetry, along the diagonal joining the two vertices where each pair of equal adjacent sides meets – this diagonal bisects the angles at those two vertices (which are generally different from each other, not equal). The kite’s one pair of equal angles sits at the other two vertices, off this axis, each the reflection of the other rather than bisected by it. For example, the kite with vertices (0,2), (1,0), (0,-1), (-1,0) has its equal-side-meeting vertices at (0,2) and (0,-1), so its symmetry axis is the vertical line through them, not a diagonal through (1,0) and (-1,0).

Angles

angles on a straight line sum to 180 degrees
angles at a point sum to 360 degrees
angles in a triangle sum to 180 degrees

For a polygon with n sides: exterior angle sum = 360 degrees always, so one exterior angle of a regular polygon = 360/n; interior angle = 180 - exterior angle. (Extended) Irregular polygons are included too: their exterior angles still sum to 360 degrees, but they are not all equal.

Regular octagon: exterior angle = 360/8 = 45 degrees
                 interior angle = 180 - 45 = 135 degrees

Circle theorems

• (Extended) Angle at the centre is twice the angle at the circumference, standing on the same arc.
• Angle in a semicircle is always 90 degrees.
• (Extended) Angles in the same segment, standing on the same arc, are equal.
• (Extended) Opposite angles of a cyclic quadrilateral sum to 180 degrees – this applies only when all four vertices lie on the circle, not to the general quadrilateral angle sum of 360 degrees.
• A tangent meets a radius at 90 degrees at the point of contact.
• (Extended) Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
• (Extended) Symmetry properties: equal chords are equidistant from the centre; the perpendicular bisector of a chord passes through the centre; the two tangents from an external point are equal in length.

Always check whether all the vertices of a shape actually lie on the circle before applying a circle theorem, and never rely on the alternate segment theorem, or any theorem, without justifying every angle it depends on (for example, that a triangle is isosceles, or two lines are parallel) from information actually given.

Exam traps

• Leaving construction arcs unmarked, since the visible method is what earns the mark.
• Converting a map scale in the wrong direction, or forgetting the final unit conversion.
• (Extended) Using the linear scale factor instead of its square (area) or cube (volume) when scaling.
• Confusing rotational symmetry with line symmetry, or assuming they must match.
• Quoting a circle theorem’s name without applying the numbers, or applying numbers without naming the theorem used.
• (Extended) Applying the general 360-degree quadrilateral angle sum to a cyclic quadrilateral instead of the specific 180-degree opposite-angles rule.

Self-test

1. State what “congruent” means, as distinct from “similar.”
2. (Extended) If the linear scale factor between two similar shapes is 3, what is the volume scale factor?
3. (Extended) State the circle theorem relating the angle at the centre to the angle at the circumference on the same arc.
4. What is the sum of the interior angles of any triangle?
5. State how many lines of symmetry a (non-rhombus) kite has.
6. Describe, in one sentence, how to construct a triangle with sides 7 cm, 5 cm and 4 cm using a ruler and compasses.

Answers: 1. Congruent shapes are identical in shape AND size; similar shapes are identical in shape only, one a scaled copy of the other. 2. 3^3 = 27. 3. The angle at the centre is twice the angle at the circumference, standing on the same arc. 4. 180 degrees. 5. One, along the diagonal joining the two vertices where the pairs of equal sides meet; it bisects the angles at those two vertices, and the pair of equal angles lies one on each side of it. 6. Draw the 7 cm side, draw an arc of radius 5 cm from one end and an arc of radius 4 cm from the other, then join the point where the arcs cross to both ends, leaving the arcs visible.