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This unit facilitates you in,
defining secant and tangent.
differentiate between secant, chord and
tangent.
stating and verifying the tangent property.
'In a circle the radius at the point of
contact is perpendicular to the tangent'
stating the converse of the above
statement.
explaining the meaning of touching
circles-externally and internally.
identifying the relation between d, R and
r in touching circles.
proving the theorems logically.
solving numerical problems and riders
based on theorems and their converse.
constructing a tangent to a circle at a
point on it.
constructing tangents to a circle from an
external point.
stating the properties of DCT and TCT.
constructing DCT and TCT to two given
circles.
measuring and verifying the length of
DCT and TCT.
Construction of tangent at a
point on the circle.
Construction of tangents
when the angle between radii
is given.
Tangents from an external
point - construction and proof
Touching circles - meaning,
construction and proof.
Common tangents - DCT and
TCT.
Construction of direct and
transverse common tangents.
15
The circle is the most
primitive and rudimentary of
all human inventions. It is the
corner stone in the foundation
of science and technology. It
is the basic tool used by
engineers, designers and the
greatest artists and architects
in the history of mankind.
The ancient Indian symbol of a circle, with a
dot in the middle, known as 'bindu', was
probably instrumental in the use of circle as a
representation of the concept zero.
Circles-Tangent
Properties
352 UNIT-15
In the previous chapter we have studied about chord properties of a circle. In this chapter
let us study some more properties of circle.
If a circle and a straight line are drawn in the same plane there are three possibilities
as shown below.
Figure 1 Figure 2 Figure 3
O
Q
R
O
Q
R
B
A
O
Q
R
P
Secant: A straight line which intersects a circle at two distinct points is called a Secant.
Tangent: A straight line which touches the circle at only one point is called a tangent.
Observe the following figures. Tangents touch the circle at only one point.
P
O
B
A
O
QC D
E
O
R
F
O
G S H
AB, CD, EF and GH are the tangents drawn to circles with centre O.
P, Q, R and S are the points of contact or points of tangency.
Point of contact: The point where a tangent touches the circle is called the point of
contact.
Observe the following examples. Objects in daily life situations which represent
circles and their tangents are given.
* Straight line RQ does not
touch the circle.
* RQ is a non-intersecting
line with respect to the
circle.
* Straight line RQ intersects
the circle at A and B.
* RQ is called a secant of the
circle.
* Straight line RQ touches
the circle at one and only
one point P.
* RQ is called the tangent to
the circle at P.
Circle-Tangent Properties 353
Example 1: Here is a pulley representing a circle.
The parts of the string AB and CD represent tangents to the pulley.
Example 2: Observe the single wheel cycle on the floor.
The wheel of the cycle represents a circle and the path on the floor where
the cycle is moving represents a tangent to the wheel.
Properties of tangents:
Now let us learn an important property of tangents to a circle.
Activity 1 : Observe the given figures and complete the following table.
O
X YP
K
L
M
N
Fig. 1 Fig. 2 Fig. 3
A
B
C
D
Fig.1 Fig. 2 Fig. 3
1. Radius
2. Tangent
3. Point of contact
From the above observations we can state that,
In any circle, the radius drawn at the point of contact is perpendicular to the
tangent.
Activity 2 : Observe the following figures, measure and write the angles in the given
table.
A
D
C
B
4. Angle between the radius
and the tangent (measure
the angle using protractor)
OPX = ...... ABC = ...... LMK = .......
Fig. 1 Fig. 2 Fig. 3
A
B
PO
P
A
B
O
O B
A P
354 UNIT-15
Answer the following questions :
1. In which figure radius OP is
perpendicular to AB?
2. What is AB called in figure 3?
3. In which figure AB is a tangent? Why?
From the activity 2, we can state that, In a circle, the perpendicular to the radius at its
non-centre end is the tangent to the circle.
This is the converse statement of tangent property stated earlier.
Construction of a tangent at a point on the circle.
Example 1 : Construct a tangent at any point P on a circle of radius 3 cm.
Example 2 : Draw a circle of radius 2 cm and construct a pair of tangents at the
non-centre end points of two radii such that the angle between the radii is 100°. Measure
the angle between the tangents.
Given: Radius of the circle = r = 2 cm; Angle between the radii = 100°
Step 1 : With O as centre draw a circle of
radius 3 cm. mark a point P on the circle,
Join OP.
Step 2 : With radius equal to OP and P
as centre draw an arc to intersect the
circle at A.
Step 3 : Mark B on the arc such that
OA = AB . With the same radius, A and B
as centre draw two intersecting arcs
Step 4 : Join PC and produce
on either sides to get the
required tangent.
Angle Fig. 1 Fig. 2 Fig. 3
1. OPA ........... ........... ...........
2. OPB ........... ........... ...........
3 cm O P
A
3 cm
O P3 cm
A B
O P3 cm
A B
C
Circle-Tangent Properties 355
Step 1 : With O as centre draw Step 2 : Construct a tangent at P.
a circle of radius 2 cm. Mark
P on it and join OP.
Step 3 : At O draw POQ =100° Step 4 : At Q construct another tangent.
Measure angle between the tangents.
Angle between the tangents is 80º.
Observe that, in a circle angle between the radii and angle between the tangents drawn
at their non-centre ends are supplementary.
O P2 cm
O P2 cm
O
P
Q
100° 80°
2 cmO
P
Q
100°
2 cm
356 UNIT-15
Example 3: Draw a chord of length 4 cm in a circle of radius 2.5cm. Construct tangents
at the ends of the chord.
Given: r = 2.5cm, length of the chord = 4cm
Step 1 : With O as centre draw a circle Step 2 : Construct a chord AB of length
of radius 2.5 cm. 4cm.
Step 3 :Join OA and OB. Step 4 : Construct tangents at A and B
to the circle.
O O
4 cm
A
B
O
4 c
m
A
B
O
4 c
m
A
B
Circle-Tangent Properties 357
Corollaries :
1. The perpendicular to the tangent at the point of contact
passes through the centre of the circle.
2. Only one tangent can be drawn to a circle at any point
on it.
3. Tangents drawn at the ends of a diameter are parallel
to each other.
O
A B
C D
P
Q
Discuss the reasons for the above three statements in the class and explain them.
EXERCISE 15.1
1. Draw a circle of radius 4 cm and construct a tangent at any point on the circle.
2. Draw a circle of diameter 7 cm and construct tangents at the ends of a diameter.
3. In a circle of radius 3.5cm draw two mutually perpendicular diameters. Construct
tangents at the ends of the diameters.
4. In a circle of radius 4.5 cm draw two radii such that the angle between them is 70°.
Construct tangents at the non-centre ends of the radii.
5. Draw a circle of radius 3 cm and construct a pair of tangents such that the angle
between them is 40°.
6. Draw a circle of radius 4.5 cm and a chord PQ of length 7cm in it. Construct a
tangent at P.
7. In a circle of radius 5 cm draw a chord of length 8 cm. Construct tangents at the
ends of the chord.
8. Draw a circle of radius 4cm and construct chord of length 6 cm in it. Draw a
perpendicular radius to the chord from the centre. Construct tangents at the ends
of the chord and the radius.
9. Draw a circle of radius 3.5 cm and construct a central angle of measure 80° and an
inscribed angle subtended by the same arc. Construct tangents at the points on the
circle. Extend tangents to intersect. What do you observe?
10. In a circle of radius 4.5cm draw two equal chords of length 5cm on either sides of
the centre. Draw tangents at the end points of the chords.
A
B
C D
O
358 UNIT-15
Tangents to a circle from a point at different positions.
1. Point on the circle
Activity 1 : Draw a circle of radius 3 cm and mark a
point P on it. Construct a tangent to it at P. Try to
construct another tangent at P. Is it possible? Why?
You have already learnt that only one tangent can be
drawn to the circle at a point on it.
2. Point inside the circle.
Activity 2: Draw a circle of radius 3cm. Mark a point P
inside the circle. Is it possible to draw a tangent to the
circle from this point?
You will find that each line drawn through this point
intersects the circle at two distinct points. All these
straight lines are secants to the circle.
3. Point outside the circle
Activity 3: Draw a circle of radius 3cm.
Mark a point P outside the circle. Draw rays
from this point towards the circle. Observe
them, how many secants can be drawn to
the circle from P?
How many tangents can be drawn to the
circle from the external point P?
From the figure, you observe that only PQ
and PR touches the circle. They are
tangents to the circle drawn from P.
Only two tangents can be drawn from an external point to a circle.
The two tangents that can be drawn from an external point to a circle have special
properties. You can discover these properties by the following activity.
Observe the figures given below and answer the following questions.
X
P
Y
A
H
C
D
E
F
GP
B
P
AB
CD
EF
GH
I
J
R
Q
O
A
B
P
Fig. 1
BA
O
Fig. 2
P
.O
Circle-Tangent Properties 359
(a) Name the tangents drawn from the external point P to the circle.
(b) Measure PA and PB. What is your conclusion?
(c) Measure AOP and BOP by using protractor.
(d) Measure APO and BPO. Where are the angles formed?
Record the measurements in the table given.
Figure I II III
1. PA = ....... AOP = ....... AOP = ......
PB = ....... BOP = ....... BOP = ......
2. PA = ....... AOP = ....... AOP = ......
PB = ....... BOP = ....... BOP = ......
After observing the data complete the following statements.
1. The tangents drawn from an external point to a circle are ...........
2. The tangents drawn from an external point to a circle subtend .......... angles at the
centre.
3. The tangents drawn from an external point to a circle are ........... inclined to the
line joining the centre and an external point.
From the above activity you find that tangents drawn from an external point to a
circle
(a) are equal
(b) subtend equal angles at the centre.
(c) are equally inclined to the line joining the centre and the external point.
We have learnt the property of tangents drawn from an external point to a circle by
practical construction and measurement. Now, let us discuss the logical proof for this
property. But, before going to logical proof, answer the following questions. Dicuss in the
class. This will help you to prove the theorem.
1. What type of angles are OPB and OQB ?
2. What are OP and OQ with respect to the circle ?
3. What type of triangles are OPB and OQB ?
4. What is OB with respect to OPB and OQB ?
5. Are OPB and OQB congruent to each other ?
6. According to which postulate the triangles OPB and OQB are congruent?
Now let us prove the theorem logically.
Q
P
OB
360 UNIT-15
Theorem: The tangents drawn from an external point to a circle
(a) are equal
(b) subtend equal angles at the centre
(c) are equally inclined to the line joining the centre and the external point
Data : A is the centre of the circle. B is an external point.
BP and BQ are the tangents.
AP , AQ and AB are joined.
To Prove: (a) BP = BQ
(b) PAB = QAB
(c) PBA = QBA
Proof : Statement Reason
In APB and AQB
AP = AQ radii of the same circle
APB = AQB = 90° Radius drawn at the point of contact isperpendicular to the tangent.
hyp AB = hyp AB Common side
APB AQB RHS Theorem
(a) PB = QB CPCT
(b)PAB = QAB
(c)PBA = QBA QED
ILLUSTRATIVE EXAMPLES
Example 1 : In the figure, a circle is inscribed in a quadrilateral ABCD in which B =
90°. If AD = 23cm, AB = 29cm and DS = 5cm find the radius of the circle.
Sol: In the figure. AB, BC, CD and DA are the tangents drawn to the circle at Q, P, S and
R respectively.
DS = DR (tangents drawn from an external point D to the circle.)
but DS = 5cm (given)
DR = 5 cm
In the fig. AD = 23 cm, (given)
AR = AD – DR = 23 – 5 = 18 cm
but AR = AQ
(tangents drawn from an external point A to the circle)
AQ = 18 cm.
Q
P
AB
A
B
C
D
P
QR
S
r
r
O
Circle-Tangent Properties 361
A
O
B
P
If AQ = 18 cm then (given AB = 29 cm)
BQ = AB – AQ = 29 – 18 = 11 cm
In quadrilateral BQOP,
B Q = BP (tangents drawn from an external point B)
O Q = OP (radii of the same circle)
QBP = QOP= 90° (given)
OQB = OPB = 90° (angle between the radius and the tangent at the
point of contact.)
BQOP is a square.
radius of the circle, OQ = 11 cm
Example 2 : In the fig. AP and BP are the tangents drawn from an external point P.
Prove that AOB and APB are supplementary.
Sol: Data : O is the centre of the circle. P is an external point. PA and PB are the tangents
drawn from an external point P.
To prove : AOB + APB = 180°
Proof : In quadrilateral AOBP,
AOB + OBP + BPA + PAO = 360°
but OBP = 900 and OAP = 90°
AOB + 900 + BPA + 900 = 360°
AOB + BPA + 1800 = 360°
AOB + BPA = 360° – 180°
AOB + BPA = 1800
AOB and BPA are supplementary.
Example 3 : In the figure, PQ and PR are the tangents to a circle with centre O.
If P = 4
5O. Find O and P.
Sol: In the figure QPR + QOR = 180°( Opposite angles are supplementary)
but, QPR = 4
5QOR
4
5QOR + QOR = 180°
4 QOR + 5 QOR
5 = 180°
9 QOR = 5 × 180°
QOR= 5 180
0
9
0100
QOR = 100°
If QOR = 100° then QPR = 180° – QOR = 180° – 100° = 80°
QPR = 80°
P
Q
O
R
362 UNIT-15
Example 4 : In the figure O is the centre of the circle. The tangents at B and D intersect
each other at point P. If AB is parallel to CD and ABC = 55° Find (i) BOD (ii) BPD.
Sol: In the figure AB || CD (given)
ABC = BCD (alternate angles)
but ABC = 55° ( given)
BCD = 55°
In the figure BOD = 2 BCD (central angle is twice
the inscribed angle.)
BOD = 2 × 55° = 110°.
but BOD + BPD = 180°
BPD = 180° – BOD = 180° – 110° = 70°
Example 5 : In the figure, show that perimeter of ABC = 2 (AP + BQ + CR).
Sol: Perimeter of ABC = AB + BC + AC
= AP + PB + BQ + QC + CR + RA
but AP = AR (tangents drawn from A to the circle)
PB = BQ (tangents drawn from B to the circle)
CQ = CR (tangents drawn from C to the circle)
Perimeter of ABC = AP + BQ + BQ + CR + CR + AP
= 2AP + 2BQ + 2CR
= 2(AP + BQ + CR)
EXERCISE 15.2
A. Numerical problems based on tangent properties.
1. In the figure PQ, PR and BC are the tangents to the circle.
BC touches the circle at X. If PQ = 7cm, find the perimeter
of PBC.
2. In the figure, if QPR = 50° find the measure of QSR.
3. Two concentric circles of radii 13cm and 5cm are drawn.
Find the length of the chord of the outer circle which
touches the inner circle.
4. In the given ABC, AB = 12cm, BC = 8 cm and AC = 10cm.
Find AF, BD and CE.
O
A B
C
P
D
A
B C
P
Q
R
P
Q
R
S O
C
A B
F
D
E
P
B
C
X
Q
R
O
P
Circle-Tangent Properties 363
A
B
C DO
A
B C
P
Q
R
OO
I
A
P
B
C2
C1
A
Q
B
PO
1
O
5. ABC is a right angled triangle, right angled at A. A circle is inscribed in it. The
lengths of the sides containing the right angle are 12cm and 5cm. Find the radius of
the circle.
6. In the given quadrilateral ABCD,BC = 38cm, QB = 27cm,
DC = 25cm and AD DC find the radius of the circle.
7. In the given figure AB = BC, ABC = 680 DA and DB
are the tangents to the circle with centre O.
Calculate the measure of (i) ACB (ii) AOB and
(iii) ADB
B. Riders based on tangent properties.
1. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB+CD=AD+BC.
2. Tangents AP and AQ are drawn to circle with centre O, from an external point A.
Prove that PAQ = 2. OPQ.
3. In the figure two circles touch each other
externally at P. AB is a direct common tangent
to these circles. Prove that
(a) tangent at P bisects AB at Q (b) APB = 90°.
4. A pair of perpendicular tangents are drawn to a circle from an external point. Prove
that length of each tangent is equal to the radius of the circle.
5. If the sides of a parallelogram touch a circle. Prove that the parallelogram is a
rhombus.
6. In the figure, if AB = AC prove that BQ = QC.
7. Circles C1 and C
2 touch internally at a point A and AB is a
chord of the circle C1 intersecting C
2 at P. Prove that AP = PB.
8. A circle is touching the side BC of ABC at P. AB and AC when produced are touching the
circle at Q and R respectively. Prove that AQ = ½ (perimeter of ABC)
A
B
D
R
Q
O
C
S
P
364 UNIT-15
Construction of tangents from an external point to a circle.
Example: Draw a circle of radius 1.5cm. Construct two tangents from a point 2.5cm away
from the centre. Measure and verify the length of the tangent.
Given r = 1.5 cm, d = 2.5 cm
Step 1: Draw a line segment AB equal to 2.5 cm.
Draw a circle of radius 1.5 cm with A as
centre.
Step 3: With MA or MB as radius draw a circle to
intersect the circle already drawn mark and
name the intersecting points as P and Q.
Step 5: Length of the tangent by measurement BP = BQ = 2 cm
Verification by calculation.
t = 2 2d r = 2 2
2.5 1.5 6.25 2.25 4 2cm
EXERCISE 15.3
1. Draw a circle of radius 6 cm and construct tangents to it from an external point
10 cm away from the centre. Measure and verify the length of the tangents.
2. Construct a pair of tangents to a circle of radius 3.5cm from a point 3.5cm away
from the circle.
3. Construct a tangent to a circle of radius 5.5cm from a point 3.5 cm away from it.
4. Draw a pair of perpendicular tangents of length 5cm to a circle.
5. Construct tangents to two concentric circles of radii 2cm and 4cm from a point 8cm
away from the centre.
A BA BM
Step 2:Draw the perpendicular
bisector of AB and mark the mid point
of AB as M.
Step 4:Join BP and BQ. BP and BQ are
the required tangents.
A BM
Q
P
A BM
Q
P
Circle-Tangent Properties 365
Touching Circles
Observe the pairs of circles given below.
A B
Fig. 1 Fig. 2 Fig. 3
C D E FP
A
B
In fig. 1, two circles with centres A and B are away from each other. They do not
have any common points,
In fig. 2, two circles with centres C and D intersect at A and B. They have two
common points, A and B.
In fig. 3, two circles with centres E and F have a common point P. They are called
touching circles.
Touching circles: Two circles having only one common point of contact are called
touching circles.
O O1
Fig. 5
O
C2
C1
Fig. 6
PAB
C2
C1
In fig. 4 two circles with centres O and O1 have no common point. They have different
centres.
In figure: 5 two circles C1 and C
2 have a common centre. But they do not have a
common point on the circle. Such circles are called concentric circles.
In figure 6, circles C1 and C
2 with centres A and B have a common point between
them. These are also touching circles.
Touching circles : Two circles having only one common point of contact are called
touching circles.
Two circles, one out side the other and having a common point of contact are called
externally touching circles (or) If the centres of the two touching circles lie on either
sides of the point of contact, then they are called externally toucing circles.
Two circles one inside the other and having a common point of contact are called
internally touching circles (or) If the centres of the circles lie on the same side of the
point of contact, then they are called internally touching circles.
A B
External touching circles
P A B
Internal touching circles
P
366 UNIT-15
Relation between the distance between centres of touching circles and their radii.
Activity 1: Observe the following figures. Measure and enter the data in the table
given below.
A B
Figure 1
P A B
Figure 2
P
Fig. AP(R) BP(r) AB(d) Relation between AP, BP
and AB i.e., R, r and d
1
2
From the above observations, you find that
(i) If two circles touch each other externally, the distance between their centres is
equal to the sum of their radii [d = R + r]
(ii) If two circles touch each other internally, the distance between their centres is
equal to the difference of their radii. [d = R – r]
Activity 2: Draw two circles of radii 5cm and 3cm whose centres are at 8cm apart.
What relation do you find between sum of the radii and distance between their
centres?
Activity 3: Draw two circles of radii 5cm and 3cm whose centres are at 2cm apart.
What relation do you find between difference of the radii and the distance between
their centres.
Property related to the centres and the point of contact of touching circles.
A BP A B
P
With reference to the above touching circles,
1. Name (a) the point of contact. (b) their centres.
2. What type of angle is APB in figure 1 and ABP in figure 2?
3. How are the points A, P, B lying in the figure 1 and A, B, P in figure 2 ? Why?
From the above acitivity we can conclude that,
If two circles touch each other, their centres and the point of contact are collinear.
Now let us prove the theorem logically.
Circle-Tangent Properties 367
Theorem : If two circles touch each other, the centres and the point of contact are
collinear.
Case 1: If two circles touch each other externally, the centres and the point of contact
are collinear.
Data : A and B are the centres of touching circles.
P is the point of contact.
To prove : A, P and B are collinear
Construction : Draw the tangent XPY.
Proof : Statement Reason
In the figure
APX = 90° .....(1) Radius drawn at the point of contact
BPX = 90° .....(2) is perpendicular to the tangent.
APX + BPX = 90° + 90° by adding (1) and (2)
APB = 180° APB is a straight angle
APB is a straight line
A, P and B are collinear. QED
Case 2 : If two circles touch each other internally the centres and the
point of contact are collinear.
Data : A and B are the centres of touching circles. P is
the point of contact.
To prove : A, B and P collinear
Construction : Draw the tangent XPY. Join AP and BP.
Proof : Statement Reason
In the figure,
APX = 90°..... (1) Radius drawn at the point of contact
BPX = 90° ..... (2) is perpendicular to the tangent.
APX = BPX = 90°
AP and BP lie on the same line.
A, B and P are collinear. QED
}
}
X
Y
A BP
X
Y
AB
P
368 UNIT-15
ILLUSTRATIVE EXAMPLES
Example 1.Three circles touch each other externally. Find the radii of the circles if
the sides of the triangle obtianed by joining the centres are 10cm, 14cm and
16cm respectively.
Sol. Circles with centres A, B and C touch each other externally at P, Q and R respectively
as shown in the fig.
Let AP = x PB = BQ = 10 – x
RC = CQ = 14 – x
but, CQ + BQ = 16
14 – x + 10 – x = 16
24 – 2x = 16
24 – 16 = 2x
2x = 8
x = 8
2 = 4
Example 2.In the figure P and Q are the centres of the circles with radii 9cm and 2cm
respectively. If PRQ = 90° and PQ = 17cm find the radius of the circle with
centre R.
Sol. Let the radius of the circle with centre R = x units.
In PQR, PQ = 17cm
PR = (x + 9) cm
QR = (x + 2) cm and PRQ = 90°
PQ2 = PR2 + QR2 (Pythagoras theorem)
172 = (x + 9)2 + (x + 2)2
289 = x2 + 81 + 18x + x2 + 4 + 4x
2x2 + 22x + 85 – 289 = 0
2x2 + 22x – 204 = 0 by 2
x2 + 11x – 102 = 0 (x + 17)(x – 6) = 0
x + 17 = 0 or x – 6 = 0
x = –17 or x = 6
radius of the circle with centre R = 6cm
PQ
R x x
2 9
AR = AP = 4cm radius of the circle
with centre A
BQ = 10 – 4 = 6cm radius of the circle
with centre B
CR = 14 – 4 = 10cm radius of the circle
with centre C
A
B
C
x xR
P
Q
Circle-Tangent Properties 369
A B P
R
Q
Example 3. In the figure circles with centres A and B touch each other internally. P is
the point of contact. Prove that AR || BQ.
Sol. Let BPQ = xº.
In PBQ,
BP = BQ (radii of the same circle)
BQP = BPQ (angles opposite to equal
sides of an isosceles le)
BQP = x° ......(1) ( BPQ = x°)
Similarly, In PAR,
AP = AR (radii of the same circle)
ARP = APR (angles opposite to equal sides of an
isosceles le)
ARP = xº ......(2) ( APR = x°
from (1) and (2)
BQP = ARP (angles which are equal to same angle
are equal)
corresponding angles are equal.
AR || BQ
Hence proved.
EXERCISE 15.4
A. Numerical problems on touching circles.
1. Three circles touch each other externally. Find the radii of the circles if the sides of
the triangle formed by joining the centres are 7cm, 8cm and 9cm respectively.
2. Three circles with centres A, B and C touch each other as shown
in the figure. If the radii of these circles are 8 cm, 3 cm and 2 cm
respectively, find the perimeter of ABC.
3. In the figure AB = 10 cm, AC = 6 cm and the radius
of the smaller circle is x cm. Find x
B. Riders based on touching circles.
1. A straight line drawn through the point of contact of two circles with centres
A and B intersect the circles at P and Q respectively. Show that AP and BQ are
parallel.
2. Two circles with centres X and Y touch each other externally at P. Two diameters
AB and CD are drawn one in each circle parallel to other. Prove that B, P and C are
collinear.
3. In circle with centre O, diameter AB and a chord AD are drawn. Another circle is
drawn with OA as diameter to cut AD at C. prove that BD = 2 OC.
A
B
C
A B
P Q
CO
x
370 UNIT-15
4. In the given figure AB = 8cm, M is the mid point of AB.
Semicircles are drawn on AB with AM and MB as diameters.
A circle with centre 'O' touches all three semicircles as
shown. Prove that the radius of this circle is 1
6 AB.
Construction of touching circles.
Example 1: Draw two circles of radii 2cm and 1cm to touch each other internally and
mark the point of contact.
Given : C1 = R = 2cm, C
2 = r = 1cm
Step 1: Find 'd'
d = R – r = 2 – 1 = 1cm
Step 2: Draw AB = 1cm, with A as centre and radius equal
to 2cm draw a circle.
Step 3: With B as centre and radius equal to 1cm draw a
circle.
Step 4: Produce AB to meet the point of contact P.
EXERCISE 15.5
1. Draw two circles of radii 5 cm and 2 cm touching externally.
2. Construct two circles of radii 4.5 cm and 2.5 cm whose centres are at 7 cm apart.
3. Draw two circles of radii 4 cm and 2.5 cm touching internally. Measure and verify
the distance between their centres.
4. Distance between the centres of two circles touching internally is 2 cm. If the
radius of one of the cirles is 4.8 cm, find the radius of the other circle and hence
draw the touching circles.
Common tangents
Study the following examples.
Example 1 : Look at the wheels of the bicycle representing the
circles, which moves along the road representing a line. The line
touches both the wheels.
Example 2 : Observe the following figures, in each case you have two circles touching
the same straight line. The straight line is a tangent to both circles. Such a tangent is
called a common tangent.
Observe the position of the circles with reference to the common tangents.
You find that both the circles lie on the same side of the tangent.
A
B
P
C1
C2
A B E
F
C
D
G
H
A
P Q
C
Ox
M D B
R
Circle-Tangent Properties 371
A
B
C
D
G H
E
F
If both the circles lie on the same side of a common tangent, then the common
tangent is called a direct common tangent (DCT).
Example 3 : Observe the following figures. In each case the straight line is a common
tangent. But observe the position of the circles. The circles lie on either side of the
common tangent.
If both the circles lie on either side of a common tangent, then the common
tangent is called a transverse common tangent (TCT).
Activity: Observe the figures, in each case identify the types and number of common
tangents.'
Sun Earth
Umbra
Penumbra
E
B
A
D
Construction of direct common tangents to two congruent circles
1. Draw direct common tangets to two circles of radii 3cm, whose centres are 8cm
apart. (construction shown here is as per suitable scale)
Given: r = 3cm = C1 = C
2, d = 8cm
Step 1: Draw AB = 8cm, wtih A and B as centres draw two circles of radii 3cm.
A B
C1 C2
Step 2: Draw two perpendicular diameters at A and B using compasses.Name the points as P, Q, R and S.
A B
P Q
R S
A B
CDE
372 UNIT-15
Step 3: Join PQ and RS. PQ and RS are the required direct common tangets.
A B
P Q
R S
Note: 1. Method of constructing DCT's is same for congruent
(a) Non-intersecting circles (d > R + r)
(b) Circles touching externally (d = R + r) and
(c) Intersecting circles (d < R + r and d > R – r)
2. Direct common tangents drawn to two congruent circles are equal and
parallel to the line joining the centres.
To construct a direct common tangent to two non-congruent circles
Example 1 :Construct a direct common tangent to two circles of radii 5cm and 2cm whose
centres are 10cm apart. Measure and verify the length of the direct common tangent.
(construction done according to suitable scale)
Given:d = 10cm, C1 = R = 5cm, C
2 = r = 2cm, C
3 = R – r = 5 – 2 = 3cm
Step 1:Draw AB = 10cm, with A and B
as centres and radius 5cm and 2cm
respectively draw circles C1 and C
2.
AB
C1
C2
Step 3: Draw bisector to intersect AB
at M. with M as centre and MA as radius
draw C4 to intersect C
3 at P and Q.
AB
C1
C2C
3
M
C4
Q
P
Step 2 :With A as centre and radius 3cm
draw C3.
AB
C1
C2C3
Step 4: Join AP and produce it to meetC
1 at S.
AB
C1
C2C
3
M
C4
Q
P
S
Circle-Tangent Properties 373
Step 6:With S as centre and radius PB drawan arc to intersect C
2 at T. Join ST and BT.
AB
C1
C2C3
M
C4
Q
P
S
T
Step 5:Draw tangent PB.
AB
C1
C2C3
M
C4
Q
P
S
Note : Similarly another DCT can beconstructed to C
1 and C
2.
Step 7: By measurement, ST = 9.5 cm
By calculation, DCT = 2 2d (R r) = 2 210 (5 2) = 100 9 = 91 = 9.5cm
Construction of transverse common tangents to two circles
Example 1 : Construct a transverse common tangent to two circles of radii 4cm and 2cm
whose centres are 10cm apart. Measure and verify the length of the transverse common
tangent. (constructions done according to suitable scale)
Step 1: d = 10 cm, C1 = R = 4 cm, C
2 = r = 2 cm, C
3 = R + r = 4 + 2= 6cm
Step 2 : Draw AB = 10cm with A and B as
centres draw circles of radii 4 cm and 2 cm
respectively. Name the circle as C1 and C
2.
Step 4:Draw a bisector to intersect AB at M.
With M as centre and MA as radius draw C4
to cut C3 at P and Q.
A B
C1C2
Step 3 : With A as centre and radius
6cm, draw C3.
A B
C1C
2
C3
A B
C3
C2
C1
M
C4
P
Q
Step 5:Join AP to intersect C1 at S.
A B
C3
C2
C1
M
C4
P
Q
S
Tips : To check the accuracy of
construction, PBTS must be a rectangle.
374 UNIT-15
Step 6 :Draw tangent PB.
Step 8 : By measurement, ST = 8cm,
By calculation, TCT = 2 2d (R r) = 2 210 (4 2) = 100 36 = 64 = 8cm
Note: (1) Method of constructing DCT's is same for non congruent.
(a) Non-intersecting circles (d > R + r)
(b) Circles touching externally (d = R + r) and
(c) Intersecting circles (d < R + r)
Method of constructing TCT's is same for congruent and non-congruent
non-intersecting circles (d > R + r)
EXERCISE 15.6
I. (A). 1. Draw two congruent circles of radii 3 cm, having their centres 10 cm apart.Draw a direct common tangent.
2. Draw two direct common tangents to two congruent circles of radii 3.5 andwhose distance between them is 3 cm.
3. Construct a direct common tangent to two externally touching circles of radii4.5 cm.
4. Draw a pair of direct common tangents to two circles of radii 2.5 cm whosecentres are at 4 cm apart.
(B). 1. Construct a direct common tangent to two circles of radii 5 cm and 2cm whosecentres are 3 cm apart.
2. Draw a direct common tangent to two internally touching circles of radii 4.5cm and 2.5 cm.
3. Construct a direct common tangent to two circles of radii 4 cm and 2 cm whosecentres are 8 cm apart. Measure and verify the length of the tangent.
4. Two circles of radii 5.5 cm and 3.5 cm touch each other externally. Draw adirect common tangent and measure its length.
5. Draw direct common tangents to two circles of radii 5 cm and 3 cm havingtheir centres 5 cm apart.
A B
C3
C2
C1
M
C4
P
Q
S
Step 7:With S as centre and radius PB drawan arc to intersect C
2 at T. Join ST.
A B
C3
C2
C1
M
C4
P
Q
S
T
Note : Similarly
another TCT can
be constructed to
C1 and C
2.
Circle-Tangent Properties 375
6. Two circles of radii 6 cm and 3 cm are at a distance of 1 cm. Draw a directcommon tangent, measure and verify its length.
II.(A). 1. Draw a transverse common tangent to two circles of radii 6 cm and 2 cm whosecentres are 8 cm apart.
2. Two circles of radii 4.5 cm and 2.5 cm touch each other externally. Draw atransverse common tangent.
3. Two circles of radii 3 cm each have their centres 6cm apart. Draw a transversecommon tangent.
(B). 1. Draw a transverse common tangent to two congruent circles of radii 2.5 cm,whose centres are at 8 cm apart.
2. Two circles of radii 3 cm and 2 cm are at a distance of 3 cm. Draw a transversecommon tangent and measure its length.
3. Draw transverse common tangents of length 8 cm to two circles of radii 4 cmand 2 cm.
4. Construct a transverse common tangent to two circles of radii 4 cm and 3 cmwhose centres are 10 cm apart. Measure and verify by calculation.
5. Construct two circles of radii 2.5 cm and 3.5 cm whose centres are 8 cm apart.Construct a transverse common tangent. Measure its length and verify bycalculation.
ANSWERS
EXERCISE 15.2
A. 1] 14 cm 2] 065QSR 3] AB = 24cm 4] AF = 7cm, BD = 5 cm, CE = 3 cm
5] 2cm 6] 14cm 7] (i) 056ACB (ii) 0112AOB (iii) 068ADBEXERCISE 15.4
A. 1] 4cm, 3cm and 5cm 2] 16cm 3] x = 2.4 cm
Tangent Properties
Tangent at a point on a circle
Tangent at the ends of the diameter
Tangents at the non-centre ends
of the radii
Tangent to a circleif angle between the
tangents is given
Tangent at the ends of the chord
PropertiesConstructions Common tangents to two circles.
DCT TCT
To two congruent circles
To two non-congruent circles
* Externally non-touching* Externally touching* Intersecting
Internally touching
Externally non-touching
Externally touching
Tangents from an external point to circle
Touching circles
Numerical Problems Riders
376 UNIT-15
Let
us s
um
mari
se t
he d
ata
regard
ing c
om
mon
tan
gen
ts. S
tudy t
he g
iven
table
.
Sl.
Cir
cle
sF
igures
Rela
tion
betw
een
Num
ber
of
Num
ber
of
Num
ber
of
No.
d, R
and r
DC
T'S
TC
T'S
CT'S
1.
Non
-in
ters
ecti
ng
d >
R +
r2,
2,
4
and n
on
-tou
ch
ing
PQ
, RS
TU
, V
W
cir
cle
s
2.
Cir
cle
s t
ou
ch
ing
d =
R +
r2,
1,
3
exte
rnally
PQ
, RS
XY
3.
Inte
rsecti
ng c
ircle
sd <
R +
r2,
-2
an
d d
> R
– r
MN
, PQ
4.
Cir
cle
s t
ou
ch
ing
d =
R –
r1,
-1
inte
rnall
yE
F
5.
Non
-con
cen
tric
d <
R –
r-
--
cir
cle
san
d d
> 0
6.
Con
cen
tric
cir
cle
sd =
0-
--
