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Mathematics · Class 10 · Chapter 29: Practical Geometry — Circles

8 min

Key points

Construction of tangents to a circle from an external point using compass and ruler only
A tangent to a circle is perpendicular to the radius at the point of contact
From any external point, exactly two tangents can be drawn to a circle
The two tangents drawn from an external point to a circle are equal in length
Construction of a circle circumscribing a triangle (circumcircle)
Construction of a circle inscribed in a triangle (incircle)
The perpendicular bisectors of the sides of a triangle meet at the circumcenter
The angle bisectors of a triangle meet at the incenter
Common tangents to two circles — external and internal tangents

Definitions

Tangent: A tangent to a circle is a straight line which touches the circle at one point only. The point where the tangent touches the circle is called the point of contact.
Circumcircle: A circle which passes through all the vertices of a triangle is called the circumcircle of the triangle.
Incircle: A circle which touches all the sides of a triangle is called the incircle of the triangle.
Circumcenter: The center of the circumcircle of a triangle is called the circumcenter. It is the point where the perpendicular bisectors of the sides meet.
Incenter: The center of the incircle of a triangle is called the incenter. It is the point where the angle bisectors of the triangle meet.

Worked example

Define tangent to a circle. Construct a tangent to a given circle from an external point P. (5 marks)

A tangent to a circle is a straight line that touches the circle at exactly one point. Construction completed using compass and ruler showing both tangents PA and PB from external point P.

Common mistakes

Students forget to prove that PA ⊥ OA in tangent construction
Confusing circumcenter with incenter — remember circumcenter uses perpendicular bisectors
Not marking the construction arcs clearly in diagrams
Forgetting that tangents from external point are equal in length
Drawing inaccurate perpendicular bisectors without proper compass technique

Quick recap

Practical geometry of circles involves precise definitions and constructions. Tangent touches circle at one point, secant intersects at two points, chord connects two points on circle. Key constructions include tangents from external points using auxiliary circles, and circles through three points using perpendicular bisectors. Always use compass and ruler for constructions.

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.