Tangents — O/L Mathematics

What must I remember about Tangents?

• Theorem 1: The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact (OP ⊥ PT).
• Theorem 2: Tangents drawn to a circle from an external point are equal in length (PA = PB), subtend equal angles at the centre, and subtend equal angles with the line joining the external point to the centre.
• Converse Theorem 1: A line drawn perpendicular to a radius at its extremity on the circle is a tangent to the circle.
• Alternate Segment Theorem: The angle between a tangent and a chord drawn through the point of contact is equal to the angle subtended by the chord in the alternate segment (∠TAB = ∠ACB).
• Cyclic quadrilateral tangent relationships: Useful in multi-step circle geometry riders and formal deductive proofs.

Test yourself (1 minute)

1. Alternate segment chord tracing

   Identify the chord meeting the tangent at point of contact; trace that chord to the circumference opposite — THAT is the equal alternate segment angle.

2. External tangents right triangle

   The line joining the external point to the centre (OP) bisects the angle between the tangents and forms two congruent right-angled triangles with the radii.

3. Geometric reason notation

   In Paper 2 proofs, cite 'tangent ⊥ radius', 'tangents from ext. point =', or 'alternate segment theorem'.

Practise Tangents questions today

Source: NIE G.C.E. (O/L) Mathematics Syllabus (Grade 11, Unit 22)

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What comes next?

Next in the syllabus

Where does Tangents fit in the syllabus?

Grade 11 · Term 3