GEOMETRY

CIRCLE THEOREMS, TRANSFORMATION AND SYMETRY

GEOMETRY

MATHS

Circle theorems

Circle theorems constitute a substantial portion of geometry marks in Paper 4 and Extended Paper 2. CIE IGCSE Mathematics examinations require both calculation and proof using these theorems.

Circle theorem 1: The angle subtended by an arc at the centre is twice the angle subtended at the circumference.

Circle theorem 2: Angles subtended by the same arc in the same segment are equal.

Circle theorem 3: The angle in a semicircle is 90°.

Circle theorem 4: Opposite angles in a cyclic quadrilateral sum to 180°.

Circle theorem 5: The angle between a tangent and a chord equals the angle in the alternate segment (alternate segment theorem).

Circle theorem 6: A tangent to a circle is perpendicular to the radius at the point of contact.

Circle theorem 7: Two tangents drawn from an external point to a circle are equal in length

Transformations

Reflection

requires specification of the mirror line (equation or description). Common mirror lines include x = k, y = k, y = x, and y = -x.

Rotation requires three pieces of information:

• Centre of rotation (coordinates)

• Angle of rotation (degrees, with direction)

• Direction (clockwise or anticlockwise)

Translation

Described by a column vector (x/y) indicating horizontal and vertical movement.

Enlargement requires:

• Centre of enlargement (coordinates)

• Scale factor (positive values enlarge or reduce; negative values also reflect through the centre)

Scale factors between 0 and 1 produce reductions

Negative scale factors produce images on the opposite side of the centre

Combination transformation

May appear in multi-part questions where students perform successive transformations or identify a single transformation equivalent to two combined operations

Symmetry and similarity

Symmetry

Line symmetry

occurs when a shape reflects onto itself across a mirror line. Regular polygons with n sides have n lines of symmetry.

Rotational symmetry

occurs when a shape rotates onto itself. The order of rotational symmetry counts how many times the shape fits onto itself during a 360° rotation. Shapes with no rotational symmetry have order 1.

Similarity problems require identifying corresponding sides and calculating with scale factors.

When shapes are similar with scale factor k:

• Corresponding lengths are in ratio k:1

• Areas are in ratio k²:1

• Volumes are in ratio k³:1

This relationship between linear and area scale factors frequently appears in examination questions.

Worked examples

Example 1: In the diagram, ABC is a triangle with AB parallel to DE. AB = 8 cm, DE = 12 cm, and AC = 10 cm. Calculate the length of AE.

Solution:

Since AB is parallel to DE, triangles CAB and CED are similar (corresponding angles are equal).

The scale factor from triangle CAB to triangle CED = 12 ÷ 8 = 1.5

Therefore CE = 10 × 1.5 = 15 cm

AE = CE - AC = 15 - 10 = 5 cm

Answer: AE = 5 cm

[This example demonstrates the standard similarity approach tested across CIE papers, worth typically 3-4 marks with method marks for identifying similar triangles and calculating scale factor correctly.]

Example 2: A circle has centre O. Points A, B and C lie on the circumference. The tangent to the circle at point A meets the line BC extended at point T. Angle BAC = 48° and angle TAB = 65°. Calculate angle AOB.

Solution:

Angle TAB = angle ACB = 65° (alternate segment theorem)

In triangle ABC: Angle ABC = 180° - 48° - 65° = 67° (angles in a triangle sum to 180°)

Angle AOB = 2 × angle ACB = 2 × 65° = 130° (angle at