CIRCLES

CIRCLES

1. DEFINITION: The collection of all the points in a plane which are at a fixed distance from a fixed point in the plane is called a circle.

   • The fixed point is called the centre of the circle.
   • The line segment joining any two points on the circumference of the circle is called a chord.
     • A segment of a circle is a region in the interior of the circle enclosed by an arc and a chord.
       • MINOR SEGMENT
       • MAJOR SEGMENT
   • The line segment joining the centre and any point on the circle is called the radius of the circle.
   • A part of the circumference of a circle is called an arc. It is two types 1)MAJOR ARC 2)MINOR ARC
   • The perimeter of the circle is called the circumference.
   • A chord which passes through the centre of a circle is called the diameter and it is the longest chord which is two times the radius.
     • A sector is a region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.
       • MAJOR SECTOR
       • MINOR SECTOR

2. Equal chords of a circle subtend equal angle at the centre.

   • Equal angles at the centre are subtended by equal chords.

3. The perpendicular from the centre of the chord bisects the chord.

   • The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

4. There is one and only one circle passing through three given non collinear points.

5. Equal chords of a circle or congruent circle are equidistant from the centre.

   • Chords equidistant from the centre of a circle are equal in length.

6. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

   • Angles in the same segment of a circle are equal.
   • Angle in a semicircle is a right angle.

7. If two arcs of a circle are congruent then their corresponding chords are equal

   • If two chords of a circle are equal then their corresponding arcs are congruent.

8. Congruent arcs of a circle subtend equal angle at the centre.

9. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment the four points lie on the circle.

   • The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree
     • If sum of a pair of opposite angles of a quadrilateral is 180 degree,

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