THEOREM: If two chords AB and CD of a circle intersect inside the circle at point P or intersect outside the circle if produced to point P then AP × BP = CP × DP TANGENT:

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Geometry – 4

CIRCLE AND THEIR PROPERTIES

THEOREM:

In a circle the angle in a semi-circle is a right angle

ACB= 900

CYCLIC QUADRILATERAL

Definition:

If all the vertices of a quadrilateral lie on a

circle, it is called a cyclic quadrilateral.

THEOREM:

The sum of the opposite angles of

a cyclic quadrilateral is 1800.

BAC + BDC = 1800

THEOREM

The exterior angle of a cyclic quadrilateral

is equal to its interior opposite angle.

∠CDE = ∠ ABC



THEOREM

The angle subtended at the center by

an arc of a circle is double the angle

which this arc subtends at the remaining

part of the circumference.

AOB = 2ACB

THEOREM

Angles in the same segment of a circle are equal.

ACB = ADB

RELATIONS AMONG TANGENT, CHORD AND SECANT

If line touches the circle at one point only then it is called a Tangent

If line connect the two point at the circle then it is called a Chord

If line starting from outside the circle intersect the circle at two points then it is

called Secant



THEOREM:

If two chords AB and CD of a circle intersect inside the circle at point P or

intersect outside the circle if produced to point P then

AP × BP = CP × DP

TANGENT:

APB is called a tangent to the circle.

The touching point P is called the point of contact.

The lengths of tangents drawn from an external point to a circle are equal:

The lengths of tangents drawn from

an external point to a circle are equal.

Tangent PQ = Tangent PR

THEOREM:

If from the point of contact of a tangent, a chord

is drawn then the angles which the chord makes

with the tangent line are equal respectively to the

angles formed in the corresponding alternate segments.

∠ DAT = ∠BCA

∠BAP = ∠BDA



THEOREM:

PA. PB = PT2

A chord and a tangents intersect externally

then the product of the lengths of the

segments of the chord is equal to the

square of the length of the tangent from

the point of contact to the point of intersection

The tangent at any point of a circle and the radius through this point are perpendicular to each other.

OP AB

Perpendicular bisector of a chord passes through the Centre.

i.e. AD = DB

OD ⊥ AB

So, O is the Centre of the circle

Equal chords of a circle are equidistant

from the Centre.

If AB = PQ and OD = OR

Chords which are equidistant from

the Centre in a circle are equal.

If OD = OR

Then, AB = PQ



If the sides of a quadrilateral touch a circle,

the sum of a pair of opposite sides is equal

to the sum of the sides of the other pair.

AB + CD = AD + BC

DIRECT COMMON TANGENT:

If r1, r2 are the radii of two circles and d is

distance between their centres, then

The length of a Direct

common tangent LM = PQ

=√𝑑2 − (𝑟1 − 𝑟2)2

TRANSVERSE COMMON TANGENT

If r1, r2 are the radii of two circles and d is

distance between their centres, then

sThe length of a Transversal

common tangent LQ = PM

=√𝑑2 − (𝑟1 + 𝑟2)2

INCIRCLE

In-radius (r) = ∆

S

Where,

∆ = Area of triangle

s = Semi perimeter

S = a+b+c

2



CIRCUMCIRCLE

OB is a Circum-radius

Circum-radius(R)

= abc

4∆

Some previous year questions

Ex: In circle with centre O two chords AB and CD intersect at P . IF ∠𝐀𝐃𝐏 =

230 and ∠𝑨𝑷𝑪 = 700 then find the ∠𝐁𝐂𝐃?

Sol: △APC,

∠APD = 1800 -700 = 1100

∠PAD = ∠PCB = 1800 –(1100 + 230) = 470

Ex: In circle O two secant PBA and PDC intersect at P and AD is diameter. IF

∠𝑫𝑨𝑷 = 400 and ∠𝑨𝑷𝑪 = 200 then find the ∠𝐃𝐁𝐂 ? 𝐒𝐒𝐂 𝐂𝐆𝐋 (𝐈)𝟐𝟎𝟏𝟑

Sol:

△ADP,

∠ADC = ∠DAP + ∠APD

= 200 +400 = 600

∠ADC = ∠ABC = 600

∠ABD = 900

∠DBC = ∠ABD - ∠ABC = 900 – 600 = 300



Ex: In circle center O, AC is chord. B is point on arc AC in such that ∠𝐎𝐂𝐀 =

200 then find ∠ 𝐀 ?

Sol:

O is center of circle so OA = OC

∠OAC = ∠OCA = 200 △APC,

∠AOC = 1800 – (200 + 200 ) = 1400

∠ADC = 700

∠ABC = 1800 –700 = 1100

Ex: A , B and C are three point on circle center O chord AB and AC make angle

900 and 1100 at center , then find the ∠𝐁𝐀𝐂 ?

Sol:

O is centre of circle OA = OC

∠OAC = ∠OCA =1

2(180 − 110) = 350

So

∠OAB = ∠OBA = 1

2( 1800 − 900) = 450

∠BAC = 450 +350 = 800



Ex: Two circle with centre P and Q intersect at B and C.A, D are points on

the circle such that A,C, D are collinear . If 𝐀𝐏𝐁 = 1300 and 𝐁𝐐𝐃 = x0

then find the value of x ?

Sol:

∠AMB = 1

2(1300) = 650

Cyclic quadrilateral MACB

∠AMB = ∠BCD = 650

∠BQD = 2∠BCD = 1300

Ex: Each of circles of equal radii with centre A and B passes through the

center of on another circle they cut at C and D then find ?

Sol:

∆ABC AB = BC = CA

∴ ∠A = ∠B = ∠C = 600

Similarly ∆ABD

∠A = ∠B = ∠D = 600

∴ ∠DBC = 60 + 60 = 1200



Ex: In Fig. P is outside the circle and secants PCA and PDB intersect the

circle at C and A, D and B respectively. If PA = 24, CA = 16 and DB = 26,

find PB.

Sol:

Let PD = x

Now, PD × PB = PC × PA

x (x + 26) = 8 × 24

x2 + 26x - 192 = 0

(x + 32) (x - 6) = 0

x = 6, positive value of x.

PB = 6 + 26 = 32.

Ex: In Fig. chords AB and CD intersect at a point P. If CP = 6 cm. CD = 9 cm.

and AB = 19 cm., what are the lengths of AP and PB ?

Sol:

CP = 6 cm, DP = CD - CP = 9 - 6 = 3 cm

Let AP = x, then PB = AB - AP = (19 – x)

Now AP . PB = CP . DP

x (19 - x) = 6 × 3 19x - x2 = 18

x2 - 19x + l8 = 0 (x - 1) ( x - 18) = 0

x = 1, 18

Hence, the lengths of AP and PB are 1 cm. and 18 cm.



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THEOREM: If two chords AB and CD of a circle intersect inside the circle at point P or intersect outside the circle if produced to point P then AP × BP = CP × DP TANGENT: