Students can go through AP 9th Class Maths Notes Chapter 9 Circles to understand and remember the concepts easily.

## Class 9 Maths Chapter 9 Notes Circles

Circle : A circle is a round simple plane figure whose boundary consists of all points that are equidistant from a fixed point.

(or)

A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.

→ That fixed point is called centre of the circle.

→ In the diagram ‘O’ is the centre. And A’ is a point on its boundary.

→ Radius : The distance between a point on the boundary of a circle and its centre is called Radius of the circle. It is denoted by ‘r’.

In the diagram \(\overline{\mathrm{OA}}\) = r.

→ Arc : The curved part between any two points on the boundary is called Arc.

In the diagram \(\overparen{P Q}\) is arc.

→ We can draw any number of radii to a circle.

→ We can show any number of arcs on the boundary of a circle.

→ Chord : The line segment joining any two points on the boundary of a circle is called chord.

In the following diagram \(\overline{\mathrm{AB}}\),\(\overline{\mathrm{PQ}}\), \(\overline{\mathrm{CD}}\) are few examples for chords.

We can draw any number of chords to a circle.

→ Diameter : The chord moving through the centre of a circle is called diameter. So we call diameter as the biggest chord. In this figure

\(\overline{\mathrm{AB}}\) is diameter

and \(\overline{\mathrm{OA}}\) → radius

\(\overline{\mathrm{OB}}\) → radius

∴ AB = OA + OB = r + r = 2r

∴ Diameter (d) = 2r

So diameter is double to its radius.

→ Major arc : The larger arc of a circle is called major arc.

In the following diagram \(\overparen{P R Q}\) is called major arc.

major arc

→ Minor arc : The smaller arc of a circle is called minor arc.

In the above figure \(\overparen{P S Q}\) is called minor arc.

→ Angle subtended by a line segment:

Let \(\overline{\mathrm{AB}}\) is a line segment and P, Q are two different points which are not on the line / line segment.

Then the angles APB, AQB are called angle subtended by a line segment.

→ Angles subtended by chord : In the following figures ‘O’ is centre, AB is chord, P, Q, R are 3 points on the circle then, ∠APQ, ∠AQB, ∠ARB are 3 angles subtended by same chord. Hence they are equal. ∠AOB is angle subtended by \(\overline{\mathrm{AB}}\) at centre.

Chords ↔ angles subtended at centre.

→ We can observe the following :

- Angles subtended by longer chords at centre are always bigger.
- Angles subtended by shorter chords at centre are always smaller.
- Angles subtended by equal chords at centre are always equal and viceversa.

→ Distance : The length of the perpendicular from a point to a line is the distance of the line from the point.

- Equal chords ↔ equal distances from centre.

It means equal chords of a circle are at equal distances from the centre of the circle.

If chords AB and CD are equal then their distances from centre also equal it means OP = OQ. - The chords of a circle which are equidistant from the centre are of equal lengths.

Angles subtended by an arc of a circle :

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

In the above diagram

i) \(\overparen{\mathrm{ABC}}\) is the corresponding arc of

chord \(\overline{\mathrm{AC}}\).

ii) \(\overparen{\mathrm{DEF}}\) DEF is the corresponding arc of chord \(\overline{\mathrm{DF}}\).

iii) \(\overparen{\mathrm{GHI}}\) is the corresponding arc of chord \(\overline{\mathrm{GI}}\).

So, if \(\overparen{\mathrm{ABC}}\) = \(\overparen{\mathrm{DEF}}\) = \(\overparen{\mathrm{GHI}}\) then

\(\overline{\mathrm{AC}}\) = \(\overline{\mathrm{DF}}\) =

\(\overline{\mathrm{GI}}\)

So, if \(\overline{\mathrm{AC}}\) = \(\overline{\mathrm{DF}}\) =

\(\overline{\mathrm{GI}}\)

\(\overparen{\mathrm{ABC}}\) = \(\overparen{\mathrm{DEF}}\) = \(\overparen{\mathrm{GHI}}\)

- Equal arcs of a circle subtends equal angles at centre.
- Angles in the same segment of a circle are equal.

That means

If the arcs \(\overparen{\mathrm{ABC}}\) = \(\overparen{\mathrm{CDE}}\) then the angles made by them at centre x°, y° also equal i.e. x° = y°.

→ Cyclic quadrilateral : A quadrilateral PQRS is called cyclic quadrilateral. If all its four vertices lie on a circle, (as shown below)

- Sum of either pair of opposite angles of a cyclic qudrilateral is 180°

i.e. if PQRS is a cyclic quadrilateral then ∠P + ∠R = ∠Q + ∠S = 180° then ABCD is a cyclic quadrilateral. - Angle in a semicircle is a right-angle.
- 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.

→ Theorem 9.1 : Equal chords of a circle subtend equal angles at the centre.

Proof : You are given two equal chords AB and CD of a circle with centre O (see Fig). You want to prove that

∠AOB = ∠COD

In triangles AOB and COD,

OA = OC (Radii of a circle)

OB = OD (Radii of a circle)

AB = CD (Given)

Therefore, ΔAOB ≅ ΔCOD (SSS rule)

This gives ∠AOB = ∠COD

(Corresponding parts of congruent triangles)

→ Theorem 9.2

Statement : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

Given : Let ‘O’ is the centre of circle

AB, PQ are two chords and

∠AOB = ∠POQ (angles at centre are equal)

Proof: Consider ΔAOB, ΔPOQ then

\(\overline{\mathrm{OA}}\) in ΔAOB = \(\overline{\mathrm{OP}}\) in ΔPOQ

(∵ OA = OP = radius)

\(\overline{\mathrm{OB}}\) in ΔAOB = \(\overline{\mathrm{OQ}}\) in ΔPOQ

(∵ OB = OQ = radius) now the included angles

∠AOB in ΔAOB = ∠POQ (given)

Hence from SAS criterion of congruency, the above two triangles are congruent.

∴ ΔAOB ≅ ΔPOQ

Then from CPCT, we can conclude AB = PQ.

Hence chords are equal.

Hence proved.

→ A collection of all points in a plane which are at a fixed distance from a fixed point in the sapie plane is called a circle. The fixed point is called the centre and the fixed distance is called the radius of the circle.

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

→ The longest of all chords which passes through the centre is called a diameter.

→ Circles with same radii are called congruent circles.

→ Circles with same centre and different radii are called concentric circles.

→ Diameter of a circle divides it into two semi-circles.

→ The part between any two points on the circle is called an arc.

→ The area enclosed by a chord and an arc is called a segment. If the arc is a minor arc then it is called the minor segment and if the arc is major arc then it is called the major segment.

→ The area enclosed by an arc and the two radii joining the end points of the arc with centre is called a sector.

→ Equal chords of a circle subtend equal angles at the centre.

→ Angles in the same segment are equal.

→ An angle in a semi circle is a right angle.

→ If the angles subtended by two chords at the centre are equal, then the chords are congruent.

→ The perpendicular from the centre of a circle to a chord bisects the chords. The converse is also true.

→ There is exactly one circle that passes through three non-collinear points.

→ The circle passing through the three vertices of a triangle is called a circumcircle.

→ Equal chords are at equal distance from the centre of the circle, conversely chords at equidistant from the centre of the circle are equal in length.

→ Angle subtended by an arc at the centre of the circle is twice the angle subtended by it at any other point on the circle.

→ If the angle subtended by an arc at a point on the remaining part of the circle is 90°, then the arc is a semi circle.

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

→ The sum of pairs of opposite angles of a cyclic quadrilateral are supplementary.