# Circles

A circle is a set of those points in a plane that are at a given constant distance from a given fixed point in the plane. The fixed point is called the centre and the constant distance is called the radius.

Area of a circle is Ï€rÂ². Circumference of a circle is 2Ï€r.

**Tangent:** A line meeting a circle in only one point is called a tangent to the circle. The point at which the tangent line meets the circle is called the point of contact.

**Secant:** A line which intersects a circle in two distinct points is called a secant line.

# Some qualities of circles are given below

1. A tangent touches a circle at only one point. A chord is any line joining any two points on the circle. When the chord pass through the centre, it becomes the diameter.

2. A tangent is perpendicular to the radius.

3. The perpendicular from the centre of a circle to a chord bisects the chord. In the figure, If ON âŠ¥ AB, then AN = NB.

# Converse

1. The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord

In the figure, If AM = MB, then OM âŠ¥ AB.

2. The perpendicular bisectors of two chords of a circle intersect at its centre. Equal chords of a circle subtend equal

angles at the centre In the figure, If AB = CD, then âˆ 1 = âˆ 2

**Converse**: If the angles subtended by two chords at the centre of a circle are equal then the chords are equal.

3. A perpendicular from the centre of the circle to the mid-point of the chord is perpendicular to the chord.

Equal chords are equidistant from the centre. The reverse is also true.

4. There is only one circle that can pass through three non-collinear points.

5. Tangents drawn from an external point are equal. The lengths of two tangents drawn from an external point to a circle are equal. If two tangents AP and AQ are drawn from a point A to a circle C, then AP = AQ

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

7. Angles in the same segment are equal In the figure, âˆ ACB and âˆ ADB are angles at the circumference, standing on the same arc, then

âˆ ACB = âˆ ADB

8. The angle in a semicircle is a right angle Explanation: In the figure given below âˆ ACB = 90Â°

**Converse:** The circle drawn with hypotenuse of a right triangle as diameter passes through its opposite vertex.

The circle drawn with the hypotenuse AB of a right triangle ACB as diameter passes through its opposite vertex C.

9. In a cyclic quadrilateral, the sum of the opposite angles is 1800. If one side of cyclic quadrilaterals produced, then the exterior angle is equal to the interior opposite angle. The quadrilateral formed by angle bisectors of a cyclical quadrilateral is also cyclic.

In the figure, the sum of the either pair of the opposite angles of a cyclic quadrilateral is 180Â° ABCD is a cycle quadrilateral, then âˆ A + âˆ C = âˆ B + âˆ D = 180Â°

** **

**Converse:** If the two angles of a pair of opposite angles of a quadrilateral are supplementary them the quadrilateral is cyclic.

If a side of a cycle quadrilateral is produced then the exterior angle is equal to the interior opposite angle In the figure, the side AB of a cycle quadrilateral ABCD is produced then âˆ 1 = âˆ 2.

10. Equal arcs make equal chords.

11. If two chords AB and CD intersect externally at P, then PA.PB = PC.PD (The triangles PAC and PBD are similar).

If two chords AB and CD intersect internally (ii) or externally (i) at a point P then PA Ã— PB = PC Ã— PD

12. When two circles touch, their centres and the point of contact are collinear. If they touch externally, the distance between their centres is equal to the sum of radii and if the circles touch internally, the distance between the centres equals the difference of the radii.

13. If from the point of contact of tangent, a chord is drawn then the angle, which the chord makes with the tangent, is equal to the angle formed by the chord in the alternate segment.

14. Area of the circle is Ï€r2.

15. If PAB is a secant to a circle intersecting the circle at A and B is a tangent segment then PA Ã— PB = PT2 Known as Tangent-Secant theorem:

16. Alternate Segment Theorem: In the figure below, if BAC is the tangent at A to a circle and if AD is any chord, then âˆ DAC = âˆ APD and âˆ PAB = âˆ PDA (Angles in alternate segment)

17. If two tangents are drawn to a circle from an external point then

(i) They subtend equal angles at the centre.

(ii) They are equally inclined to the segment, joining the centre to that point.

In a circle C, A is a point outside it and AP and AQ are the tangents drawn to the circle

Then, âˆ 1 = âˆ 2 and âˆ 3 = âˆ 4

18. If a circle touches all the four sides of a quadrilateral then the sum of opposite pair of sides are equal If ABCD is a circumscribed quadrilateral, then AB + CD = AD + BC

19. The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. If ABCD is a cyclic quadrilateral in which AP, BP, CR and DR are the bisectors of âˆ A, âˆ B, âˆ C and âˆ D respectively then quadrilateral PQRS is also cyclic.

20. A cyclic trapezium is isosceles and its diagonals are equal If ABC cyclic trapezium such that AB || DC, then AD = BC and AC = BD

**Converse:** An isosceles trapezium is always cyclic.