Circle Class 9 ||Maths|| Chapter 9 NCERT Notes

Circle ||Maths|| Chapter 8 NCERT Notes

1. Basic Definitions

  • Circle: A circle is the set of all points in a plane that are equidistant from a fixed point. The fixed point is called the center of the circle, and the constant distance from the center to any point on the circle is called the radius.

  • Radius: A line segment joining the center of the circle to any point on the circle.

  • Diameter: A chord that passes through the center of the circle. It is the longest chord of the circle and is equal to twice the radius.

    Diameter=2×Radius

  • Chord: A line segment joining any two points on the circle. The diameter is the longest chord of the circle.

  • Arc: A part of the circumference of a circle.

  • Circumference: The total length of the boundary of the circle.

  • Sector: A region enclosed by two radii of the circle and an arc.

  • Segment: A region of the circle enclosed by a chord and the corresponding arc.

2. Terms Related to Circles

  • Secant: A line that intersects a circle at two points.

  • Tangent: A line that touches the circle at exactly one point. The point where a tangent touches the circle is called the point of contact. The tangent is perpendicular to the radius at the point of contact.

  • Concentric Circles: Two or more circles having the same center but different radii.

  • Cyclic Quadrilateral: A quadrilateral inscribed in a circle such that all its vertices lie on the circle.

3. Angle Subtended by a Chord at a Point

  • A chord subtends an angle at a point on the circle.
  • The angle subtended by a chord at the center of the circle is called the central angle.
  • The angle subtended by a chord at any other point on the circle is called an angle subtended on the circle.

4. Theorems on Circles

Theorem 1: Equal Chords Have Equal Angles

  • Equal chords subtend equal angles at the center of the circle.

Theorem 2: The Perpendicular from the Center to a Chord Bisects the Chord

  • If a perpendicular is drawn from the center of the circle to a chord, it bisects the chord. This means the perpendicular divides the chord into two equal parts.

Theorem 3: Angles in the Same Segment of a Circle are Equal

  • Angles subtended by the same arc at different points on the circle are equal.

Theorem 4: Angle Subtended by the Diameter is a Right Angle

  • The angle subtended by the diameter of a circle at any point on the circle is 90°. This is a special case of Theorem 3.

Theorem 5: Cyclic Quadrilateral

  • The sum of the opposite angles of a cyclic quadrilateral is 180°.

Theorem 6: Tangent to a Circle

  • The tangent to a circle is perpendicular to the radius drawn to the point of contact.

Theorem 7: Lengths of Tangents from an External Point are Equal

  • From any point outside the circle, the lengths of the two tangents drawn to the circle are equal.

5. Tangent to a Circle

  • A tangent is a straight line that touches the circle at exactly one point.
  • Important properties:
    1. The tangent is perpendicular to the radius at the point of contact.
    2. From any point outside a circle, two tangents can be drawn to the circle, and both are of equal length.

6. Cyclic Quadrilaterals

  • A quadrilateral is said to be cyclic if all its vertices lie on the circumference of a circle.

    Properties of a Cyclic Quadrilateral:

    • The sum of the opposite angles of a cyclic quadrilateral is 180°.
    • Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

7. Arc and Angle Relationships

  1. Angle at the Center vs. Angle on the Circle:

    • The angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the remaining part of the circle.

  2. Angle in a Semi-Circle:

    • The angle subtended by a diameter on the circle is a right angle (90°).

Summary:

  • A circle is defined by its center and radius, with various related terms such as chords, tangents, and secants.
  • Important properties and theorems of circles include the perpendicularity of tangents and radii, equal tangents from an external point, and the angle relationships in cyclic quadrilaterals.
  • Understanding these theorems helps in solving problems related to circles, tangents, and cyclic figures.