3 circle 1

CIRCLE
Circles passing through one,
two, three points
Circles touching each other
Inscribed angle and
intercepted arc
Secant and tangent
Arc of a circle
Cyclic
quadrilateral

Circle is a set of all points in
the plane which are
equidistant from a given
point, called the center of a
circle.
Radius of circle R is the distance from the centre О of the
circle to any point on the circle.
Diameter of circle D is a segment that connects two points on circle and passes through its center.
The distance from the centre of the circle to the secant (chord) is always smaller than the radius.
CIRCLE

CIRCLE
PROPERTIES OF THE CHORD OF A CIRCLE
Chords of equal length are equidistant from the
centre of the circle.
if chords AB = CD, then
ON = OK
A perpendicular from the centre of
the circle to the chord, divides chord
in two equal parts (bisects the chord)
if OC ┴ AB, then
AC = BC

CIRCLE
Circles passing through one, two or three points
Infinite circles pass through one point.
Infinite circles pass through two distinct points.
There is a unique circle passing through three non-collinear
points.
No circle can pass through 3 collinear points.

CIRCLE
SECANT AND TANGENT
Not a single point is common in line l and circle with centre A.
Point P is common to both, line m and circle with centre B. Here, line m is called a
tangent of the circle and point P is called the point of contact.
Q and R are intersecting points of line n and the circle. Line n is called a
secant of the circle .
A line perpendicular to a radius at its point on the circle
is a tangent to the circle.

CIRCLE
TANGENT
A line perpendicular to a radius at its
point on the circle is a tangent to the
circle.
Tangent segments drawn from an
external point to a circle are
congruent.

CIRCLE
In the adjoining figure,
circle with centre D touches the sides
of ∠ACB at A and B. If
∠ ACB = 52°, find measure of ∠ ADB.
Solution : The sum of all angles of a quadrilateral is 360°.
∠ ACB + ∠ CAD + ∠ CBD + ∠ ADB = 360°
52° + 90° + 90° + ∠ ADB = 360° Tangent theorem
∠ ADB + 232° = 360°
∠ ADB = 360° - 232°
= 128°

CIRCLE
1. m∠ CAB = 900 tangent theorem
2. D(C,A) = 6 tangent is at a
perpendicular distance (radius)
from the centre of a circle
3. △ CAB is an isosceles right angled
triangle, Using 45 -45 -60 theorem
4. AC =
1
2
X BC
6 2 = BC.
5. 𝐴𝐶 = AB as △ CAB is an isosceles
right angled triangle, m∠ABC = 450
In the adjoining figure the radius of a circle with centre C is 6
cm, line AB is a tangent at A. Answer the following questions.
What is the measure of ∠CAB ? Why ?
What is the distance of point C from
line AB? Why ? d(A,B)= 6 cm, find d(B,C).
What is the measure of ∠ ABC ? Why ?

CIRCLE
1. In △ MOR, OM =
1
2
X OR,
m ∠ OMR = 900 TANGENT THM
Using 30-60-90 thm, m∠ ORM =300
𝑚∠ MOR =600
RM =
3
2
X OR
=
3
2
x 10 = 5 3
1. m∠ MRO =300
2. m∠ MRN =600
In the adjoining figure, O is the centre
of the circle. From point R,
seg RM and seg RN are tangent
segments touching the circle at
M and N. If (OR) = 10 cm and radius of the circle
= 5 cm, then
What is the length of each tangent segment ?
What is the measure of ∠ MRO ? (3) What is
the measure of ∠ MRN ?

CIRCLE
In △ MOR and △ NOR,
m ∠ OMR = m ∠ ONR= 900 TANGENT THM
side OM ≅ side ON radii of the same circle
side RM ≅ side RN tangent segments
△ MOR ≅ △ NOR S-A-S TEST
∠ ORM ≅ ∠ ORN C.A.C.T
∠ MOR ≅ ∠ NOR C.A.C.T.
Seg OR bisects ∠ MRN and ∠ MON
Seg RM and seg RN are tangent
segments of a circle with centre O. Prove that seg OR
bisects ∠MRN as well as ∠ MON.

CIRCLE
What is the distance between two parallel
tangents of a circle having radius
4.5 cm ? Justify your answer.
d(A,B) = d(A,O) + d (B,O)
d(A,B) = 4.5 + 4.5 (radii of the same circle)
= 9 cm
(parallel tangents are at a distance of a rdiameter ).

CIRCLE
The circles with centre R and S touch the line l in point T. So they
are two touching circles with l as common tangent. They are touching
externally.
The circles with centre M and N touch each other internally and line p
is their common tangent.
The circles shown are called externally touching circles.
The circles shown are called internally touching circles
P
TOUCHING CIRCLES

CIRCLE
The point of contact of the touching circles
lies on the line joining their centres.
The distance between the centres of the circles
touching externally is equal to the sum of their
radii. d(R,S) = R1 + R2
touching internally is equal to the difference of
their radii. d(N,M) = R1 - R2 (R1>R2)
TOUCHING CIRCLES

CIRCLE
Two circles having radii 3.5 cm and 4.8 cm
touch each other internally. Find the
distance between their centres.
Two circles of radii 5.5 cm and 4.2 cm
touch each other externally. Find the
distance between their centres
R1 = 4.8 cm, R2 = 3.5 cm
Circles touch each other internally
hence
R1 = 5.5 cm, R2 = 4.2 cm
Circles touch each other externally
hence
d(N,M) = R1 - R2
= 4.8 - 3.5
= 1.3 cm
d(R,S) = R1 + R2
= 5.5 + 4.2
= 9.7 cm

CIRCLE
If radii of two circles are 4 cm and 2.8 cm. Draw figure of these
circles touching each other - (i) externally (ii) internally.

CIRCLE
PROVE THAT
1.in∆APR, AP ≅ PR Radii of the same circle .
∴ m∠ PAR ≅ m∠ PRA property of an isosceles triangle
in∆QRB, RQ ≅ ---- Radii of the same circle .
∴ m∠ QRB ≅ ------- property of an isosceles triangle
2.m∠ PRA = -------- v. Opposite angles
3.∴ m∠ PAR = ---------- from 1 and 2
4.∴ seg AP ∥ seg BQ from 3, alternate angle property
5.in ∆APR and ∆𝑅𝑄B
∠ PAR ≅ ∠ -------- from 3
∠ PRA ≅ ∠ -------- FROM 2
6.∴ ∆APR ∼ ∆𝑅𝑄B BY AA TEST
m∠ PAR = 350 given
∴m∠ ------- =350 from 5
and m∠ QRB = 350 from 1
∴m∠ RQB = 180 – 35 – 35 = 1100
(sum of the 3 angles of a triangle is
1800)
Ans on next slide

CIRCLE
PROVE THAT
1.in∆APR, AP ≅ PR Radii of the same circle .
∴ m∠ PAR ≅ m∠ PRA property of an isosceles triangle
in∆QRB, RQ ≅ QB Radii of the same circle .
∴ m∠ QRB ≅ m∠ QBR property of an isosceles triangle
2.m∠ PRA = m∠ QRB v. Opposite angles
3.∴ m∠ PAR = m∠ QBR from 1 and 2
4.∴ seg AP ∥ seg BQ from 3, alternate angle property
5.in ∆APR and ∆𝑅𝑄B
∠ PAR ≅ ∠ QBR from 3
∠ PRA ≅ ∠ QRB FROM 2
6.∴ ∆APR ∼ ∆𝑅𝑄B BY AA TEST
m∠ PAR = 350 given
∴m∠ QBR=350 from 5
and m∠ QRB = 350 from 1
∴m∠ RQB = 180 – 35 – 35 = 1100
(sum of the 3 angles of a triangle is
1800)

CIRCLE
The circles with centres A and B touch each other
at E. Line l is a common tangent which touches
the circles at C and D respectively. Find the length
of seg CD if the radii of the circles are 4 cm, 6 cm
CD is a common tangent.
∴ AC ⊥ CD, BD ⊥ CD tangent thm
Construct AX ∥ CD ∴
ACDX is a rectangle and XB = 6-4 = 2 cm
in∆AXB, m∠ AXB = 900 (corresponding angle)
AB = 6+4 =10
(USE PYTHAGORAS THEOREM to find AX
which is then equal to CD)

CIRCLE
Measure of a minor arc is equal to the measure of its
corresponding central angle.
Measure of major arc = 360° - measure of corresponding minor arc.
Measure of a semi circular arc, that is of a semi circle is 180°.
Measure of a complete circle is 360°. 310
80

CIRCLE
Relation between the arcs and chords of a circle
ABCD IS A RECTANGLE.
AD = BC ∴ arc AD = arc BC
AB = CD ∴ arc AB = arc CD

CIRCLE
ABCD is a square
AB = BC = CD = DA
∴ arc AB = arc BC = arc CD = arc DA
M(arc AB) = m( arc BC )= 𝑚 (arc CD)= m( arc DA)
=
360
4
=90
ABC is an equilateral triangle
AB = BC = CA
∴ arc AB = arc BC = arc CA
m(arc AB) = m( arc BC )= 𝑚 (arc CA)
=
360
3
= 120

CIRCLE
CENTRAL ANGLE
Central angle is the angle, the apex of which is the
centre of circle.
.In a circle, the degree measure of an arc is equal to the measure
of the central angle that intercepts the arc.

CIRCLE
In figure, points G, D, E, F
are concyclic points of a circle with
centre C.
∠ ECF = 70°, m(arc DGF) = 200°
find m(arc DE) and m(arc DEF).
E F
∠ ECF = m (arc EF)
70° = m (arc EF)
m(arc DE) = 360 - [ m (arc DGF + m (arc EF)]
m(arc DE) = 360 – (200° + 70° ) = 90°
m(arc DEF) = m(arc DE) + m (arc EF)
= 90 + 70 = 160°

CIRCLE-REVISION
A perpendicular from the centre of the circle to the chord,
divides chord in two equal parts (bisects the chord
Chords of equal length are equidistant from the centre of the
circle.
A line perpendicular to a radius at its point on the circle is a
tangent to the circle.
Tangent segments drawn from an external point to
a circle are congruent.
d(R,S) = R1 + R2
d(N,M) = R1 - R2 (R1>R2
Measure of major arc = 360° - measure of corresponding minor arc

CIRCLE-REVISION
A perpendicular from the centre of the circle to the chord,
divides chord in two equal parts (bisects the chord)
Chords of equal length are equidistant from the centre of the
circle. if chords AB = CD, then ON = OK
A line perpendicular to a radius at its point on the circle is a
tangent to the circle.
Tangent segments drawn from an external point to a circle are
congruent.
Name the chord and radius shown
If AC = 5 cm, state the distance of chord from the centre
If BD = 10 cm , what is BC?
Chord BD, radius AB, DIST 5 cm, BC =5

CIRCLE-REVISION
If OZ =4, OX = 4.5, OY=4, Name the CONGRUENT chords.
EX = 4 , FIND FX, EF.
•Name the tangents shown in the figure.
At what point is line m tangent to the given circle.
Name the tangents segments from point B
If CN = 4 cm, what is DC? Why?
If AX = 4 cm, BX= 5 cm and CN = 2cm find AD, CD, BN and perimeter of triangle ABC.
Chord AB =chord CD, fx = 4, ef = 8, line n and line m, at point B is line m a tangent,
BX & BN, CD =4 (TANGENT SEGMENTS ARE CONGRUENT, AD = 4, CD = 2, BN = 5, PERIMETER =22 CM

CIRCLE-REVISION
Find the measures of the central angle and/or
corresponding arc /arcs marked with ? mark.
Answer on next slide

CIRCLE-REVISION
m∠𝑄𝑂𝑃 = 𝑚 𝑎𝑟𝑐 𝑄𝑆𝑃 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒)
m∠𝑄𝑂𝑃 = 50°
M(arc QSP) + m(arc QRP) =360 (Sum of the measure of all the arcs of a circle is 360 degree)
5 0 + m(arc QRP) =360
m(arc QRP) = 310

CIRCLE
INSCRIBED ANGLE OF A CIRCLE
Inscribed angle is the angle inside the circle, the
apex of which lies on the circle
In each of the above figures, arc lies
in the interior of an angle.
The arc is called an intercepted arc
by an inscribed angle.

CIRCLE
In each of the above figures,
two arcs are in the interior of
an angle.
Those arcs are called an
intercepted arcs by an
inscribed angle.
X
X X
X
X
X

CIRCLE
All inscribed angles, based on one arc is equal (one end of the chord).
Inscribed angle will be a 90°, if it is based
on the diameter of the circle.
Any inscribed angle is always equal to half the central angle,
based on the same arc
Inscribed angle is the angle inside the circle, the
apex of which lies on the circle

CIRCLE
CENTRAL ANGLE
m∠ AOB = m (arc AXB) m∠ ACB =
1
2
m (arc AXB)

CIRCLE-REVISION
Find the measures of the inscribed angle
or intersepting arc marked with ? mark.

CIRCLE-REVISION
m∠𝐵𝐴𝐶 = 1
2
m(arc BDC)
40 =
1
2
m(arc BDC)

CIRCLE-REVISION
Find the measures of the inscribed angle or
intersepting arc marked with ? mark

CIRCLE
∠ AEC = [m(arc AC) + m(arc DB)]
∠ BED = [m(arc BD) - m(arc AC)]
1
2
1
2
1
2
∠ ACB = [m(arc AXB)
1
2
∠ CEB = [m(arc AD) + m(arc CB)]

CIRCLE
The measure of an inscribed angle is half the measure
of the arc intercepted by it.
Angles inscribed in the same arc are congruent.
Angle inscribed in a semicircle is a right angle.
Opposite angles of a cyclic quadrilateral are supplementary.
∠ BED = [m(arc BD) - m(arc AC)]
∠ AEC = [m(arc AC) + m(arc DB)] and ∠ CEB = [m(arc AD) + m(arc CB)]
1
2
1
2
1
2

CIRCLE-REVISION
Find measure of the angle marked

CIRCLE-REVISION
m∠𝐸𝐴𝐶 = 1
2
[80 -20) }
m∠𝐸𝐴𝐶 = 1
2
[m(arc EGC) - m(arc DHB) =
1
2
m(arc BDC)

CIRCLE-REVISION
m∠𝑄𝑂𝑅 = 1
2
[20+40]
m∠𝑄𝑂𝑅 = 1
2
[m(arc QR) + m(arc WE)

CIRCLE-REVISION
m∠ 𝑇𝑂𝑈 =
1
2
m (arc TYU)
m∠ 𝑄𝑃𝐸 = m (arc QWE)

CIRCLE-REVISION
m∠𝑄𝑃𝑊 =
1
2
[ m (arc QTW) + m (arc RTE)] m∠ 𝐷𝐹𝐺 =
1
2
[ m (arc SH) - m(arc DG)]
m∠ WPE =
1
2
[ m (arc WE) + m (arc QR)]

CIRCLE-REVISION
m ∠𝐵𝑃𝑉 =
m ∠𝑋𝑃𝐶 =
m ∠𝐵𝑂𝑉 =
m ∠𝑋𝑂𝐶 =
m ∠𝐵𝑍𝑉 =
m ∠𝐵𝑋𝑉 =
m ∠𝐵𝐶𝑉 =
Ans: 100, 30, 65, 65, 35, 50, 50

If two inscribed angles based on a chord are
located on either side of it, the sum of the
angles is 180°.
α + β = 180°
CIRCLE – CYCLIC QUADRILATERAL
1.Arc BCD is intercepted by ∠ BAD
Arc BAD is intercepted by ∠ BCD
2.∠ BAD =
1
2
Arc BCD
∠ BCD =
1
2
Arc BAD
3.∠ BAD + ∠ BCD =
1
2
(Arc BCD + Arc BAD )
4. ∠ BAD + ∠ BCD =
1
2
X 3600
5. ∠ BAD + ∠ BCD = 1800
If two points on a given line subtend equal
angles at two different points which lie on
the same side of the line, then those four
points are concyclic

CIRCLE-REVISION
Find measure of the angle marked Answer on next slide

CIRCLE-REVISION
m∠ QWE + m∠ QRE = 180 (OPP ANGLES OF A CYCLIC QUADRILATERAL ARE SUPPLEMENTARY ANGLES
m∠ QWE + 80 = 180

CIRCLE
In D QRS is an equilateral triangle. Prove that,
arc RS @ arc QS @ arc QR
m(arc QRS) = 240
D QRS is an equilateral triangle
Chord QR = chord RS = chord SQ
Arc QR = Arc RS = Arc SQ =
360
3
Arc QR = Arc RS = Arc SQ = 120 °,
Arc QRS = m(arc QR + arc RS) = 120 +120 = 240 °,

In fig , chord AB ≅ chord CD,
Prove that,
arc AC ≅ arc BD
chord AB ≅ chord CD,
Arc AB ≅ arc CD
Arc AC + arc CB ≅ arc CB + arc BD
arc AC ≅ arc BD
CIRCLE

In figure, in a circle with centre O,
length of chord AB is equal to the
radius of the circle. Find measure of
each of the following.1) ∠ AOB
(2) ∠ ACB (3) arc AB(4) arc ACB.
AO = OB = AB (GIVEN)
m ∠ AOB = 60° (property of an equilateral triangle
Arc AB = 60°
𝑚 ∠ ACB =
𝟏
𝟐
m(arc AB) 𝑚 ∠ ACB =
𝟏
𝟐
x 60 =30°
M(arc ACB) = 360 - m(arc AB) = 360 -60 = 300°
CIRCLE

In figure, c PQRS is cyclic.
side PQ @ side RQ.  PSR = 110°, Find-
1.measure of  PQR
2.m(arc PQR)
3.m(arc QR)
4.measure of  PRQ
c PQRS is cyclic
m  PSR + m PQR = 180
110 + m PQR =180
m PQR =70°
m  PSR =
𝟏
𝟐
m(arc PQR)
110 X 2 = m(arc PQR)
PQ = RQ
RQ = PQ = 220/2 =100
m  PRQ =
𝟏
𝟐
m(arc PQ)
= 220/2 = 100
CIRCLE

Quadrilateral ∠MRPN is cyclic, ∠ R = (5x - 13)°, ∠ N = (4x + 4)°. Find
measures of ∠ R and ∠ N.
∠R + ∠ N = 180
5X – 13 + 4X + 4 = 180
9 X -9 =180
9X = 189
X = 189/9
X = 21
∠ R = (5x - 13)
= 5X21 -13 = 105 -13= 92°
∠ N = (4x + 4)°.
= 4 X 21 +4 = 88°
CIRCLE

In figure, seg RS is a diameter of the circle with
centre O. Point T lies in the exterior of the
circle.
Prove that
∠ RTS is an acute angle.
∠ ∠
∠
∠ RXS = 90 (ANGLE INSCRIBED IN A SEMICIRCLE)
∠ XTS is a remote interior angle of ∠ RXS
BY EXTERIOR ANGLE THEOREM, ∠ RXS > ∠ XTS
m ∠ XTS < 90
Hence, ∠ xTS = ∠ RTS is an acute angle
CIRCLE

Prove that, any rectangle is a cyclic quadrilateral.
m ∠A =m ∠B = m ∠C = m ∠D = 90
m ∠A + m ∠C =180
HENCE, any rectangle is a cyclic
quadrilateral
CIRCLE

CIRCLE
In figure, altitudes YZ and XT of
∠ WXY intersect at P. Prove that,
1. Quadrilateral WZPT is cyclic.
2. Points X, Z, T, Y are concyclic
M ∠ 𝑊𝑍𝑃 + M ∠ 𝑊𝑇𝑃 = 180 property of linear pair of angles
Using property of cyclic quadrilateral,
Quadrilateral WZPT is cyclic. (opp angles are supplementary
∠ 𝑋𝑍𝑌 = ∠ 𝑋𝑇𝑌 Given
Using property of cyclic quadrilateral,
Points X, Z, T, Y are concyclic (If two points on a given line subtend equal angles at
two different points which lie on the same side of the
line, then those four points are concyclic)

CIRCLE
In figure, m(arc NS) = 125°,
m(arc EF) = 37°, find the measure  NMS.
M
∠ NMS = [125°- 37°)]
1
2
∠ NMS = [m(arc NS) - m(arc EF)]
1
2
∠ NMS = [88°]
∠ NMS = 44°
1
2

CIRCLE
A E
In figure, chords AC and DE
intersect at B. If  ABE = 108°,
m(arc AE) = 95°, find m(arc DC)
∠ ABE = [m(arc AE) + m(arc DC)]
108° X 2 = [95° + m(arc DC]
216-95 = m(arc DC)
131° = m(arc DC)
1
2

CIRCLE - REVISION
TANGENT
A line perpendicular to a radius at its
point on the circle is a tangent to the
circle.
Tangent segments drawn from an
external point to a circle are
congruent.

The point of contact of the touching circles
lies on the line joining their centres.
touching externally is equal to the sum of their
radii. d(R,S) = R1 + R2
touching internally is equal to the difference of
their radii. d(N,M) = R1 - R2 (R1>R2)
TOUCHING CIRCLES
CIRCLE - REVISION

Relation between the arcs and chords of a circle
ABCD IS A RECTANGLE.
CIRCLE - REVISION

If two inscribed angles based on a chord are
located on either side of it, the sum of the
angles is 180°.
α + β = 180°
CIRCLE – CYCLIC QUADRILATERAL
If two points on a given line subtend equal
angles at two different points which lie on
the same side of the line, then those four
points are concyclic

Measure of major arc = 360° - measure of corresponding minor arc.
Measure of a semi circular arc, that is of a semi circle is 180°.
Measure of a complete circle is 360°. 310
80
CIRCLE - REVISION

CENTRAL ANGLE
m∠ AOB = m (arc AXB) m∠ ACB =
1
2
m (arc AXB)
CIRCLE - REVISION

1 From the information given in the figure, find the measure of ∠AEC. a. 42° b. 30°
c. 36° d. 72°
∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB). a. 65°
b. 130° c. 295° d. 230°
If AB∥CD in the given figure, O is the centre of the circle. If ∠BAD =60°, then
∠ADC is equal to a. 30 b. 45 c. 60 d. 120
Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between
their centers ? a. 4.4 cm b. 8.8 cm c. 2.2 cm d. 8.8 or 2.2 cm
Two circles intersect each other such that each circle passes through the centre of the other.
If the distance between their centres is 12, what is the radius of each circle ? a. 6 cm b.
12 cm c. 24 cm d. can’t say
Hint is given on next 4 slides

∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB). a. 65°
b. 130° c. 295° d. 230°
m∠ ACB =
1
2
m (arc AXB)

1 From the information given in the figure, find the measure of ∠AEC. a. 42° b. 30°
c. 36° d. 72°
𝐻𝐼𝑁𝑇: ∠ AEC = [m(arc AC) + m(arc DB)] and ∠ CEB = [m(arc AD) + m(arc CB)]
1
2
1
2

If AB∥CD in the given figure, O is the centre of the circle. If ∠BAD =60°, then
∠ADC is equal to a. 30 b. 45 c. 60 d. 120
AB∥CD. ∠ADC = ∠BAD ( )

CIRCLE-REVISION
Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What
is the distance between their centers ? a. 4.4 cmb. 8.8 cm c. 2.2 cm
d. 8.8 or 2.2 cm
d(R,S) = R1 + R2 d(N,M) = R1 - R2)

A, B, C are any points on the circle with centre O.
Write the names of all arcs formed due to these points.
If m arc(BC = 110 and m arc(AB) = 125, find measures of all remaining arcs

CIRCLE-REVISION
Two circles intersect each other such that each circle passes through the
centre of the other. If the distance between their centres is 12, what is the
radius of each circle ? a. 6 cm b. 12 cmc. 24 cm d. can’t say
Select the appropriate figure to get answer
Ans: 12 cm

CIRLE-REVISION
In the following figure,O is the centre of the circle. ∠ABC is inscribed in arc
ABC and ∠ABC = 65°. Find the measure of ∠ AOC.
m∠ ACB =
1
2
m (arc AXB)
m∠ AOB = m (arc AXB)

CIRCLE-REVISION
∠
2
∠
1.In the figure m (arc LN) = 110°, m (arc PQ) = 50° find ∠LMN.
LMN = 1 [m (arc LN) - [ ]
2
∠ BED =
1
2
[m(arc BD) - m(arc AC)]

m∠ AOB = m (arc AXB)
m∠ ACB =
1
2
m (arc AXB)
For reference

Opposite angles of a cyclic
quadrilateral are supplementary

𝐻𝐼𝑁𝑇: ∠ interior =
1
2
[m(arc 1) + m(arc 2)]
∠ exterior =
1
2
[m(arc 1) - m(arc 2)]

CIRCLE-REVISION
In the figure, m (arc APC) = 100° and ∠BAC = 70°.
Find i. ∠ABC ii. m (arc BQC).
m∠ ACB =
1
2
m (arc AXB)

CIRCLE-REVISION
In the figure, ∠ABC = 80°. Find m (arc APC).

CIRCLE-REVISION
In fig, ∆QRS is an equilateral triangle. Prove that,
arc RS ≅ arc QS ≅ arc QR
m(arc QRS) = 240

CIRCLE-REVISION
Prove that, (i) arc PQ ≅ arc SR
(ii) arc SPQ ≅ PQR

CIRCLE-REVISION
In the figure along side, side PQ ≅ side PR, ∠PRQ = 77° m(arc QR) = ?

In figure, □PQRS is cyclic. side PQ ≅ side RQ. ∠PSR =
1100, Find Measure of ∠PQR, m(arc PQR), m(arc QR)
Measure of ∠PRQ

m∠ABC is 10 more than m ∠ADC.
𝐹𝑖𝑛𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 ∠ABC and ∠ADC
CIRCLE-REVISION
Let m∠ADC = x, m∠ABC = x + 10.
Frame equation to get answer

m∠SPQ ∶ m ∠SRQ = 5: 4.
𝐹𝑖𝑛𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 m∠SPQ and m ∠SRQ
CIRCLE-REVISION
Let m∠SPQ = 5x, m∠SRQ = 4x
Frame equation to get answer

Altitudes YZ and XT of ∆WXY intersect
at P. Prove that, □WZPT is cyclic.
CIRCLE-REVISION
Use property of a cyclic quadrilateral to prove .

In the adjoining figure circles with centres X and Y touch each other at point Z. A
secant passing through Z intersects the circles at points A and B respectively. Prove
that, radius XA || radius YB.
CIRCLE-REVISION

Altitudes YZ and XT of ∆WXY intersect at P.
Prove that, Points X, Z, T, Y are concyclic.
CIRCLE-REVISION

If M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = 6√3 then find -
(1) Radius of the circle.
(2) Measures of ∠K and ∠M.
CIRCLE-REVISION

In the figure, points G, D, E, F are concyclic
points of circle with centre C.
∠ECF = 70°, m(arc DGF) = 200° Find m(arc DE)
and m(arc DEF).
CIRCLE-REVISION

Line l touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the
following.
What is d(O, P) = ? Why ?
If d(O, Q) = 8 cm, where does the point Q lie ?
If d(O,R) = 15 cm, How many locations of point R are line on line l ? At what distance
will each of them be from point P ?
CIRCLE-REVISION

In figure, in a circle with centre O, length of
chord AB is equal to the radius of the circle.
Find measure of each of the following.
∠AOB ii) ∠ACB iii) arc AB iv) arc ACB.
CIRCLE-REVISION

In the following figure,O is the centre of the circle. ∠ABC is
inscribed in arc ABC and ∠ABC = 65°.
find the measure of ∠ AOC.
CIRCLE-REVISION

In the figure, □ ABCD is a cyclic quadrilateral. Seg AB is a
diameter. If ∠ADC =120°, find measure of ∠BAC.
CIRCLE-REVISION

In seg AB is a diameter of a circle with centre O. The bisector of ∠ACB
intersects the circle at point D. Prove that, seg AD ≅ seg BD.
CIRCLE-REVISION

In the following figure 'O' is the centre of the circle.
∠AOB = 110°, m (arc AC) = 45°.
m (arc AXB) =
∠COB = ____
m (arc CAB) =
m (arc AYB) =
CIRCLE-REVISION

In chord EF || chord GH. Prove that, chord EG ≅ chord FH.
CIRCLE-REVISION

If two circles intersect each other at points S and R. Their
common tangent PQ touches the circle at points P, Q.
Prove that, ∠PRQ + ∠PSQ = 180°
CIRCLE-REVISION

In the adjoining figure, O is the centre of the circle. From point R, seg RM
and seg RN are tangent Segments touching the circle at M and N. If (OR) =
10 cm and radius of the circle = 5 cm, then
What is the length of each tangent segment ?
What is the measure of ∠ MRO ? (3) What is the measure of ∠ MRN ?
CIRCLE-REVISION

In fig chord AB = chord CD, Prove that, arc AC = arc BD
CIRCLE-REVISION

In figure, chords PQ and RS intersect at T.
Find m(arc SQ) if m∠ STQ = 58°, m ∠ PSR = 24°.
2
P
P
R
CIRCLE-REVISION

3 circle 1

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