Chapter 10 - Circles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 PDF Download

CHORD PROPERTIES OF A CIRCLE

CIRCLE
A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre and the constant distance is called the radius of the circle. The given figure consists of a circle with centre O and radius equal to r units.

 TERMS AND FACTS RELATED TO CIRCLES
Radius : A line segment joining the centre and a point on the circle is called its radius, generally denoted by r.
The plural of radius is radii.

In the figure, OA, OB and OC are the radii of a circle.

Circumference : The perimeter of a circle is called its circumference.

Circumference = 2πr

Position of a Point With Respect To a Circle
Let us consider a circle with centre O and radius r. A point P is said to lie.
(i) inside the circle, if OP < r. 
(ii) on the circle, if OP = r.
(iii) outside the circle, if OP > r.

In the figure of a circle with centre O and radius r, Z

(i) The points A, O, B lie inside the circle;
(ii) The points P, Q, R lie on the circle;
(iii) The points X, Y, Z lie outside the circle.

Interior and Exterior of a Circle
The region consisting of all those points which lie inside a circle,is called the interior of the circle.
The region consisting of all those points which lie outside a circle, is called the exterior of the circle.

Circular Region or Circular Disc
The region consisting of all those points which are either on the circle or lie inside the circle, is called the circular region.

Chord : A line segment joining any two points on a circle is called a chord of the circle.
In the figure, PQ, RS and AB are the chords of a circle with centre O.

Diameter : A chord of the circle passing through the centre of a circle is called its diameter.
In the figure, AOB is a diameter of a circle with centre O.

Diameter = 2 × Radius

Properties :

(i) Diameter is the largest chord of a circle.
(ii) All diameters of a circle are equal in length.

Secant : A line which intersects a circle in two distinct points is called a secant of the circle.
In the figure, the line  cuts the circle in two points C and D. So,  is a secant of the circle.

 Tangent :

A line that intersects the circle in exactly one point is called a tangent of the circle.
The point at which the tangent intersects the circle is called its point of contact.
In the figure, SPT is a tangent at the point P of the circle with centre O.
Clearly, P is the point of contact of the tangent with the circle.

Facts About Tangents :
(i) No tangent can be drawn to a circle through a point inside the circle:
(ii) One and only one tangent can be drawn to a circle at a point on the circle.
(iii) Two tangents can be drawn to a circle from a point outside it.
In the adjoining figure, PT₁ and PT₂ are the tangents to the circle from point P.

Touching Circles : Two circles are said to touch each other if and only if they have one and only one point
in common. Two circles may touch externally [Fig. (i)] or internally [Fig. (ii)].

The common point is called the point of contact, and the line joining their centres is called the line of centres.
A line touching the two circles is called a common tangent.
Thus, in the above figure, P is the point of contact, AB is the line of centres and PT is a common tangent.

Direct Common Tangents : A common tangent to two circles is called a direct common tangent if both the
circles lie on the same side of it. In the figure, AB and CD are two direct common tangents.

Transverse Common Tangents : A common tangent to two circles is called a transverse common tangent
if the circles lie on its opposite sides.
In the figure, PQ and RS are two transverse common tangents.

Arc : A continuous piece of a circle is called an arc of the circle.

Let P and Q be any two points on a circle with centre O.

Then, clearly the whole circle has been divided into two pieces,

namely arc PAQ and arc QBP, to be denoted by PAQ and QBP respectively.

We may denote them by PQ and QP respectively.

Minor and Major Arc : An arc less than one-half of the whole arc of a circle is called a minor arc, and an arc greater than one-half of the whole arc of a circle is called a major arc of the circle.

Thus, in the above figure, PQ is a minor arc, while QP is a major arc.

Central Angle : An angle subtended by an arc at the centre of a circle is called its central angle.
In the given figure, central angle of PQ = POQ.

Degree Measure of An Arc :
Let PQ be an arc of a circle with centre O.
If POQ = θ°, we say that the degree measure of PQ is θ° and we write, m(PQ ) = θ°. If m(PQ ) = θ°, then
m(QP ) = (360 – θ)°. Degree measure of a circle is 360°.

Congruent Arcs : Two arcs AB and CD are said to be congruent, if they have same degree measure.
AB  CD ⇔ m( AB ) = m( CD ) ⇔ AOB = COD.

Semi-Circle : A diameter divides a circle into two equal arcs. Each of these two arcs is called a semi-circle.
The degree measure of a semi-circle is 180°. In the given figure of a circle with centre O, ABC as well as
ADC is a semi-circle.

Congruent Circles : Two circles of equal radii are said to be congruent.
Concentric Circles : Circles having same centre but different radii are called concentric circles.

Concyclic Points : The points, which lie on the circumference of the same circle, are called concyclic points.
In the adjoining figure, points A, B, C and D lie on the same circle and hence, they are concyclic.

Segment : A segment is a part of a circular region bounded by an arc and a chord, including the arc and
the chord. The segment containing the minor arc is called a minor segment, while the other one is a major segment.
The centre of the circle lies in the major segment.

Alternate Segments of a Circle : The minor and major segments of a circle are called alternate segments of each other.

Sector of a Circle : The part of the plane region enclosed by an arc of a circle and its two bounding radii
is called a sector of the circle.
Thus, the region OABO is the sector of a circle with centre O.

Hence, the chords AB and CD are equidistant from the centre O.

Quadrant : One-fourth of a circular disc is called a quadrant.

Cyclic Quadrilateral : If all the four vertices of a quadrilateral lie on a circle, then such a quadrilateral is
called a cyclic quadrilateral.

If four points lie on a circle, they are said to be concyclic.
We also say that quad. ABCD is inscribed in a circle with centre O.

Theorem-1 : Equal chords of a circle subtend equal angles at the centre.
Given : A circle with centre O in which chord PQ = chords RS. 

Hence proved
 

Converse of above theorem : If the angles sutended by the chords at the centre (of a circle) are
equal, then the chords are equal.

Given : A circle with centre O. Chord PQ and RS subtend equal angles at the centre of the circle.

Hence proved

Theorem 2 : The perpendicular from the centre of a circle to chord bisects the chord.

Given : AB is a chord of a circle with centre O.
To prove : LA = LB.

Construction : Join OA and OB. 

Converse of above theorem : The straight line drawn from the centre of a circle to bisect a chord,
 is perpendicular to the chord.

Given : AB is chord of a circle with centre O and OL bisects AB.

Proof :

Theorem 3. Prove that one and only one circle, passing through three non-collinear points.
 Given : Three non-collinear points A, B, C.
To prove : One and only one circle can be drawn, passing through A, B, and C.

Construction : Join AB and BC. Draw the perpendicular bisectors of AB and BC. Let these perpendicular bisector intersect meeting at a point O.

Proof :

Hence, one and only circle can be drawn through three non-collinear points A, B and C.

Theorem 4. Equal chords of a circle are equidistant from the centre.

Given : A circle with centre O in which chord AB = chord CD;

Hence, the chords AB and CD are equidistant from the centre O.

Converse of above Theorem : Chords of a circle that are equidistant from the centre of the circle,
 are equal.

Given : AB and CD are two chords of a circle with centre O;

Hence proved.

Ex.1 Determine the length of a chord which is at a distance of 6 cm from the centre of a circle of
 radius 10 cm.

Sol. Let AB be a chord of a circle with centre O and radius 10 cm.

Join OB.
In ΔBCO, We have
OC = 6 cm

⇒ AB = BC + BC [Since the perpendicular from the centre to a chord bisects the chord  AC = BC]

Ex.2 A chord of length 30 cm is drawn in a circle of radius 17 cm. Find its distance from the centre
 of the circle.

Sol. Chord AB = 30 cm, radius OA = 17 cm

Since, perpendicular drawn from the centre of circle to a chord, bisect the chord.

Hence, the distance of the chord from the centre is 8 cm.

Ex.3 In figure, O is the centre of the circle of radius 15 cm. , AB = 18 cm and CD = 22 cm. Find PQ.

Sol. Given AB and CD be two parallel chords of a circle with centre O.

so pointsO, Q and P are collinear.
Join OA and OC.
We have, AB = 18 cm, CD = 22 cm and OA = OC = 15 cm                      [Radii of the circle]
Since the perpendicular drawn from centre of circle to a chord bisect the chord. Therefore

Now, in right triangle AOP and COQ.

Ex.4 In the figure, the diameter CD of a circle with centre O is perpendicular to the chord AB. If AB = 12 cm and CE = 3 cm, find the radius of the circle.

Sol. Join OA.

Since, perpendicular drawn from the centre of circle to a chord, bisects the chord,

Let r be the radius of the circle. Then, OA = OC = r cm.

OE = (OC – CE) = (r – 3) cm.

In right-angled ΔOEA, we have

                 [By Pythagoras Theorem]
     

Ex.5 AB and CD are two parallel chords of a circle of length 24 cm and 10 cm respectively and lie
 on the same side of its centre O. If the distance between the chords is 7 cm, find the radius
 of the circle.

Sol. 

Ex.6 PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm.
 If PQ = 16 cm and RS = 12 cm, find the distance between PQ and RS, if they lie.

(i) on the same side of the centre O.

(ii) on opposite of the centre O.

Sol.

(i) Draw the perpendicular bisector OL and OM of PQ and RS respectively.

           [By Pythagoras Theorem]

  [ The perpendicular drawn from the centre of a circle to a chord bisects the chord]

In right triangle OMR,
OR² = OM² + RM²                        [By pythagoras Theorem]
 [ The perpendicular drawn from the centre of a circle to a chord bisects the chord]

Hence, the distance between PQ and RS, if they lie on the same side of the centre O, is 2 cm.

(ii) Draw the perpendicular bisector OL and OM of PQ and RS respectively.

                    [By Pythagoras Theorem]

   [ The perpendicular drawn from the centre of a circle to a chord bisects the chord]

In right triangle OMR,
                    [By Phthagoras Theorem]
 [ The perpendicular drawn from the centre of a circle to a chord bisects the chord]

⇒ OM² = (10)² – (6)² = (10 – 6) (10 + 6) = (4) (16) = 64

Hence, the distance between PQ and RS, if they lie on the opposite side of the centre O, is 14 cm.

Ex.7 AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on opposite sides of the centre and the distance between them is 17 cm, find the radius of the circle.

Sol.

Let O be the centre of the circle and let its radius be r cm.

Since AB CD, it follows that the points O, L, M are collinear and therefore,

        [By Pythagoras Theorem]

Ex.8 In figure, AB = CB and O is the centre of the circle. Prove that BO bisects ABC.

Sol. Given : In figure, AB = CB and O is the centre of the circle.


Hence proved

Ex.9 Two circle with centre A and B intersect at C and D. Prove that

Sol. Given : Two circles with centres A and B intersect at C and D.



Hence proved

Ex.10 In the adjoining figure, AB and AC are two equal chords of a circle with centre O. Show that O lies on the bisector of BAC.

Sol. Given : AB and AC are equal chords of a circle with centre O