Class 10 Mathematics – Circle: Tangent and Chord

1. Circle Basics

• A circle is a set of points in a plane that are equidistant from a fixed point called the center.

• The distance from the center to any point on the circle is called the radius.

• A line joining two points on the circle is called a chord.

2. Chord of a Circle

• A chord is a line segment joining any two points on the circle.

• Properties of a chord:

  • The longest chord is the diameter.

  • Equal chords are equidistant from the center.

  • A line joining the center and perpendicular to a chord bisects the chord.

3. Tangent to a Circle

• A tangent is a line that touches the circle at exactly one point.

• The point where it touches the circle is called the point of contact.

• Properties of a tangent:

  • A tangent is perpendicular to the radius at the point of contact.

  • A circle can have many tangents from a point outside the circle.

  • A tangent does not cross the circle.

4. Important Relationships Between Tangent and Chord

• A tangent drawn at one end of a chord forms equal angles with the chord.

• The perpendicular from the center to a chord always bisects the chord.

• Tangents help in constructing triangles, quadrilaterals, and other geometric shapes involving circles.

5. Applications of Tangent and Chord

• Engineering and design: Roads, bridges, and circular arcs.

• Geometric constructions: Constructing shapes with circles.

• Problem-solving: Verifying circle properties, lengths, and angles.

• Real life: Wheels, roundabouts, and circular tracks use tangent and chord principles.

6. Key Points to Remember

• Chord: Connects two points on a circle; diameter is the longest chord.

• Tangent: Touches the circle at exactly one point; perpendicular to radius.

• Equal chords: Equidistant from the center.

• Perpendicular from center to chord: Bisects the chord.

• Tangent properties are useful in constructions and proofs.