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Mathematics · Class 10 · Chapter 26: Tangent to a Circle

8 min

Key points

A tangent to a circle is a straight line which touches the circle at one and only one point
The point where tangent touches the circle is called the point of contact or point of tangency
A tangent to a circle is perpendicular to the radius drawn to the point of contact
Two tangents drawn from an external point to a circle are equal in length
The angle between a tangent and a chord drawn from the point of contact equals the angle in the alternate segment
From any external point, only two tangents can be drawn to a circle
The tangent at any point on a circle is perpendicular to the diameter passing through that point
Common tangents to two circles can be internal or external depending on their position

Definitions

Tangent to a Circle: A tangent to a circle is a straight line which touches the circle at one and only one point.
Point of Contact: The point where a tangent touches the circle is called the point of contact or point of tangency.
External Point: A point which lies outside the circle is called an external point.
Common Tangent: A straight line which is tangent to two or more circles is called a common tangent.
Alternate Segment: The two segments of a circle formed by a chord are called alternate segments.

Worked example

Define tangent to a circle. Prove that tangent at any point of a circle is perpendicular to the radius through the point of contact. (6 marks)

Definition: A tangent to a circle is a straight line which touches the circle at one and only one point. Proof: Let AB be tangent to circle with center O at point P. Take any point Q on AB other than P. Then OQ passes through interior of circle, so OQ > radius OP. Since P gives shortest distance from O to line AB, therefore OP ⊥ AB. Hence proved.

Common mistakes

Students often forget that tangent is perpendicular to radius at point of contact
Confusing point of contact with center of circle in diagrams
Not mentioning RHS congruency rule when proving tangent length equality
Drawing tangents from internal point instead of external point
Forgetting to write 'given', 'to prove', and 'construction' in geometry proofs

Quick recap

Tangent is a line touching circle at exactly one point. Key theorem: tangent is always perpendicular to radius at point of contact. From external point, two tangents can be drawn and they are equal in length. Remember exact textbook definitions and draw clear labeled diagrams.

Generated by AI from Sindh Textbook Board material; review state: ai_unreviewed. MCQs, past-paper references and examiner notes are held back until a teacher has checked them.