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Geometry: circles (theorems, tangents, chords) — KCSE Mathematics

KCSE Mathematics · 108 practice questions · 4 syllabus objectives

34 easy37 medium37 hard

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

State and apply circle theorems: angle at centre = 2× angle at circumference, angles in same segment, cyclic quadrilateral, tangent-radius, alternate segment

Apply chord properties: perpendicular from centre bisects chord; equal chords are equidistant from centre

Calculate lengths of tangents, chords and arcs; find the area of sectors and segments of a circle

Geometry: circles (theorems, tangents, chords)

Sample Questions

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1
easySHORT ANSWER2 marks

In a circle, a chord XY is bisected by a line from the centre O. State the relationship between the line and the chord. (2 marks)

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Part (a) — 2 marks
The line from centre O is perpendicular to chord XY (1 mk)
The line bisects chord XY into two equal segments (1 mk)
2
easySHORT ANSWER3 marks

Calculate the length of a tangent drawn from a point 10 cm away from the centre of a circle with a radius of 6 cm. (3 marks)

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Part (a) — 3 marks
Use the formula: length of tangent = √(distance from centre² - radius²) (1 mk)
Substitute values: length of tangent = √(10² - 6²) (1 mk)
Calculate the final numerical value: length of tangent = √(100 - 36) = √64 = 8 cm (1 mk)
3
easySHORT ANSWER3 marks

In a circle, chord PQ is bisected by a line drawn from the centre O at point R. Describe how this situation illustrates the theorem that a perpendicular from the centre bisects the chord. (3 marks)

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Part (a) — 3 marks
The line OR is perpendicular to chord PQ. (1 mk)
This creates two right triangles, OPR and OQR, which are congruent. (1 mk)
Hence, PR = RQ, proving that the chord is bisected. (1 mk)
4

Identify two properties of tangents drawn from an external point to a circle. (2 marks)

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Master Geometry: circles (theorems, tangents, chords) for KCSE

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