Circles

1. In a circle, perpendicular from the centre to a chord bisects the chord. 2. In a circle, the line joining the mid-point of a chord to the centre is perpendicular to the chord. 3. Equal chords of a circle are equidistant from the centre.

4. Chords equidistant from the centre are equal. 5. Equal chords of a circle subtend equal angles at the centre. 6. Chords of a circle which subtend equal angles at the centre are equal. 7. Angle subtended by an arc at the centre is double the angle subtended by it on the remaining part of the circle. 8. The angle subtended by an arc at the centre is called the central angle of the

arc and the arc is called the intercepted arc corresponding to the central angle. 9. A quadrilateral all four of whose vertices lie on a circle is called a cyclic quadrilateral and the four vertices are called concyclic points. 10. Angle in a semicircle is a right angle. 11. Angles in the same segment of a circle are equal. 12. Opposite angles of a cyclic quadrilateral are supplementary.

IN A CIRCLE, PERPENDICULAR FROM THE CENTRE TO A CHORD BISECTS THE CHORD AB = AM + MB OM AB O A

M

B

EQUAL CHORDS OF A CIRCLE ARE EQUIDISTANT FROM THE CENTRE C A O.

N

M

AB = CD OM = ON

B

D

EQUAL CHORDS OF A CIRCLE SUBTEND EQUAL ANGLES AT THE CENTRE C A O

.

AB = CD AOB= COD

D B

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 C AOB = 2

ACB

O .

A

B

.X

ANGLE SUBTENDED BY AN ARC AT ANY POINT ON THE CIRCLE IS HALF THE ANGLE SUBTENDED BY IT AT CENTRE ACB = AOB/2 C O. A . X

B

A QUADRILATERAL ALL THE FOUR VERTICES OF WHICH LIE ON A CIRCLE IS CALLED A CYCLIC QUADRILATERAL D C

B A

OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ARE SUPPLEMENTARY D

D+ B=

A+

C

C

100°

B + D = 180° or 80° + 100° = 180° And A + C = 180° or 100° + 80° = 180°

80° A

B