144 MATHEMATICS
10
10.1 Introduction
You have studied in Class IX that a circle is a collection of all points in a plane
which are at a constant distance (radius) from a fixed point (centre). You have
also studied various terms related to a circle like chord, segment, sector, arc etc.
Let us now examine the different situations that can arise when a circle and a line
are given in a plane.
So, let us consider a circle and a line PQ. There can be three possibilities given
in Fig. 10.1 below:
Fig. 10.1
In Fig. 10.1 (i), the line PQ and the circle have no common point. In this case,
PQ is called a non-intersecting line with respect to the circle. In Fig. 10.1 (ii), there
are two common points
A and B that the line PQ and the circle have. In this case, we
call the line PQ a secant of the circle. In Fig. 10.1 (iii), there is only one point A which
is common to the line PQ and the circle. In this case, the line is called a tangent to the
circle.
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You might have seen a pulley fitted over a well which is used
in taking out water from the well. Look at Fig. 10.2. Here the rope
on both sides of the pulley, if considered as a ray, is like a tangent
to the circle representing the pulley.
Is there any position of the line with respect to the circle
other than the types given above? You can see that there cannot
be any other type of position of the line with respect to the circle.
In this chapter, we will study about the existence of the tangents
to a circle and also study some of their properties.
10.2 Tangent to a Circle
In the previous section, you have seen that a tangent* to a circle is a line that
intersects the circle at only one point.
To understand the existence of the tangent to a circle at a point, let us perform
the following activities:
Activity 1 : T
ake a circular wire and attach a straight wire AB at a point P of the
circular wire so that it can rotate about the point P in a plane. Put the system on a table
and gently rotate the wire AB about the point P to get different positions of the straight
wire [see Fig. 10.3(i)].
In various positions, the wire intersects the
circular wire at P and at another point Q
1
or Q
2
or
Q
3
, etc. In one position, you will see that it will
intersect the circle at the point P only (see position
AB of AB). This shows that a tangent exists at
the point P of the circle. On rotating further, you
can observe that in all other positions of AB, it will
intersect the circle at P and at another point, say R
1
or R
2
or R
3
,
etc. So, you can observe that there is
only one tangent at a point of the circle.
While doing activity above, you must have observed that as the position AB
moves towards the position A B, the common point, say Q
1
, of the line AB and the
circle gradually comes nearer and nearer to the common point P. Ultimately, it coincides
with the point P in the position AB of AB. Again note, what happens if ‘AB’ is
rotated rightwards about P? The common point R
3
gradually comes nearer and nearer
to P and ultimately coincides with P. So, what we see is:
The tangent to a circle is a special case of the secant, when the two end
points of its corresponding chord coincide.
Fig. 10.3 (i)
Fig. 10.2
*The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was
introduced by the Danish mathematician Thomas Fineke in 1583.
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146 MATHEMATICS
Activity 2 : On a paper, draw a circle and a
secant PQ of the circle. Draw various lines
parallel to the secant on both sides of it. You
will find that after some steps, the length of
the chord cut by the lines will gradually
decrease, i.e., the two points of intersection of
the line and the circle are coming closer and
closer [see Fig. 10.3(ii)]. In one case, it
becomes zero on one side of the secant and in
another case, it becomes zero on the other side
of the secant. See the positions PQ and PQ
of the secant in Fig. 10.3 (ii). These are the
tangents to the circle parallel to the given secant
PQ. This also helps you to see that there cannot
be more than two tangents parallel to a given
secant.
This activity also establishes, what you must have observed, while doing
Activity 1, namely, a tangent is the secant when both of the end points of the
corresponding chord coincide.
The common point of the tangent and the circle is called the point of contact
[the point A in Fig. 10.1 (iii)]and the tangent is said to touch the circle at the
common point.
Now look around you. Have you seen a bicycle
or a cart moving? Look at its wheels. All the spokes
of a wheel are along its radii. Now note the position
of the wheel with respect to its movement on the
ground. Do you see any tangent anywhere?
(See Fig. 10.4). In fact, the wheel moves along a line
which is a tangent to the circle representing the wheel.
Also, notice that in all positions, the radius through
the point of contact with the ground appears to be at
right angles to the tangent (see Fig. 10.4). We shall
now prove this property of the tangent.
Theorem 10.1 : The tangent at any point of a circle is perpendicular to the
radius through the point of contact.
Proof : We are given a circle with centre O and a tangent XY to the circle at a
point P. W
e need to prove that OP is perpendicular to XY.
Fig. 10.4
Fig. 10.3 (ii)
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CIRCLES 147
Take a point Q on XY other than P and join OQ (see Fig. 10.5).
The point Q must lie outside the circle.
(Why? Note that if Q lies inside the circle, XY
will become a secant and not a tangent to the
circle). Therefore, OQ is longer than the radius
OP of the circle. That is,
OQ > OP.
Since this happens for every point on the
line XY except the point P, OP is the
shortest of all the distances of the point O to the
points of XY. So OP is perpendicular to XY.
(as shown in Theorem A1.7.)
Remarks
1. By theorem above, we can also conclude that at any point on a circle there can be
one and only one tangent.
2. The line containing the radius through the point of contact is also sometimes called
the ‘normal’ to the circle at the point.
EXERCISE 10.1
1. How many tangents can a circle have?
2. Fill in the blanks :
(i) A tangent to a circle intersects it in point (s).
(ii) A line intersecting a circle in two points is called a .
(iii) A circle can have parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called .
3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at
a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D)
119
cm.
4. Draw a circle and two lines parallel to a given line such that one is a tangent and the
other, a secant to the circle.
10.3 Number of Tangents from a Point on a Circle
To get an idea of the number of tangents from a point on a circle, let us perform the
following activity:
Fig. 10.5
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148 MATHEMATICS
Activity 3 : Draw a circle on a paper. Take a
point P inside it. Can you draw a tangent to the
circle through this point? You will find that all
the lines through this point intersect the circle in
two points. So, it is not possible to draw any
tangent to a circle through a point inside it
[see Fig. 10.6 (i)].
Next take a point P on the circle and draw
tangents through this point. You have already
observed that there is only one tangent to the
circle at such a point [see Fig. 10.6 (ii)].
Finally, take a point P outside the circle and
try to draw tangents to the circle from this point.
What do you observe? You will find that you
can draw exactly two tangents to the circle
through this point [see Fig. 10.6 (iii)].
We can summarise these facts as follows:
Case 1 : There is no tangent to a circle passing
through a point lying inside the circle.
Case 2 : There is one and only one tangent to a
circle passing through a point lying on the circle.
Case 3 : There are exactly two tangents to a
circle through a point lying outside the circle.
In Fig. 10.6 (iii), T
1
and T
2
are the points of
contact of the tangents PT
1
and PT
2
respectively.
The length of the segment of the tangent
from the external point P and the point of contact
with the circle is called the length of the tangent
from the point P to the circle.
Note that in Fig. 10.6 (iii), PT
1
and PT
2
are the lengths of the tangents from P to
the circle. The lengths PT
1
and PT
2
have a common property. Can you find this?
Measure PT
1
and PT
2
. Are these equal? In fact, this is always so. Let us give a proof
of this fact in the following theorem.
(i)
(ii)
Fig. 10.6
(iii)
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Theorem 10.2 : The lengths of tangents drawn
from an external point to a circle are equal.
Proof : We are given a circle with centre O, a
point P lying outside the circle and two tangents
PQ, PR on the circle from P (see Fig. 10.7). We
are required to prove that PQ = PR.
For this, we join OP, OQ and OR. Then
∠ OQP and ∠ ORP are right angles, because
these are angles between the radii and tangents,
and according to Theorem 10.1 they are right
angles. Now in right triangles OQP and ORP,
OQ = OR (Radii of the same circle)
OP = OP (Common)
Therefore, ∆ OQP ≅ ∆ ORP (RHS)
This gives PQ = PR (CPCT)
Remarks
1. The theorem can also be proved by using the Pythagoras Theorem as follows:
PQ
2
= OP
2
– OQ
2
= OP
2
– OR
2
= PR
2
(As OQ = OR)
which gives PQ = PR.
2. Note also that ∠ OPQ = ∠ OPR. Therefore, OP is the angle bisector of ∠ QPR,
i.e., the centre lies on the bisector of the angle between the two tangents.
Let us take some examples.
Example 1 : Prove that in two concentric circles,
the chord of the larger circle, which touches the
smaller circle, is bisected at the point of contact.
Solution : We are given two concentric circles
C
1
and C
2
with centre O and a chord AB of the
larger circle C
1
which touches the smaller circle
C
2
at the point P (see Fig. 10.8). We need to prove
that AP = BP.
Let us join OP. Then, AB is a tangent to C
2
at P
and OP is its radius. Therefore, by Theorem 10.1,
OP ⊥ AB
Fig. 10.7
Fig. 10.8
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150 MATHEMATICS
Now AB is a chord of the circle C
1
and OP AB. Therefore, OP is the bisector of the
chord AB, as the perpendicular from the centre bisects the chord,
i.e., AP = BP
Example 2 : Two tangents TP and TQ are drawn
to a circle with centre O from an external point T.
Prove that PTQ = 2 OPQ.
Solution : We are given a circle with centre O,
an external point T and two tangents TP and TQ
to the circle, where P, Q are the points of contact
(see Fig. 10.9). We need to prove that
PTQ = 2
OPQ
Let PTQ =
Now, by Theorem 10.2, TP = TQ. So, TPQ is an isosceles triangle.
Therefore, TPQ = TQP =
11
(180° ) 90°
22
Also, by Theorem 10.1, OPT = 90°
So, OPQ = OPT – TPQ =
1
90° 90° –
2
=
11
PTQ
22
This gives PTQ = 2 OPQ
Example 3 : PQ is a chord of length 8 cm of a
circle of radius 5 cm. The tangents at P and Q
intersect at a point T (see Fig. 10.10). Find the
length TP.
Solution : Join OT. Let it intersect PQ at the
point R. Then TPQ is isosceles and TO is the
angle bisector of PTQ. So, OT PQ
and therefore, OT bisects PQ which gives
PR = RQ = 4 cm.
Also, OR =
2222
OP PR 5 4 cm 3c
m
.
Fig. 10.9
Fig. 10.10
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Now, TPR + RPO = 90° = TPR + PTR (Why?)
So, RPO = PTR
Therefore, right triangle TRP is similar to the right triangle PRO by AA similarity.
This gives
TP
PO
=
RP
RO
, i.e.,
TP
5
=
4
3
or TP =
20
3
cm.
Note : TP can also be found by using the Pythagoras Theorem, as follows:
Let TP = x and TR = y. Then
x
2
= y
2
+ 16 (Taking right PRT) (1)
x
2
+ 5
2
=(y + 3)
2
(Taking right OPT) (2)
Subtracting (1) from (2), we get
25 = 6y – 7 or y =
32 16
63
Therefore, x
2
=
2
16 16 16 25
16 (16 9)
39 9
[From (1)]
or x =
20
3
EXERCISE 10.2
In Q.1 to 3, choose the correct option and give justification.
1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from
the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm
(C) 15 cm (D) 24.5 cm
2. In Fig. 10.11, if TP and TQ are the two tangents
to a circle with centre O so that POQ = 110°,
then PTQ is equal to
(A) 60° (B) 70°
(C) 80° (D) 90°
3. If tangents PA and PB from a point P to a circle with centre O are inclined to each other
at angle of 80°, then POA is equal to
(A) 50° (B) 60°
(C) 70° (D) 80°
Fig. 10.11
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152 MATHEMATICS
4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
5. Prove that the perpendicular at the point of contact to the tangent to a circle passes
through the centre.
6. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4
cm. Find the radius of the circle.
7. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the
larger circle which touches the smaller circle.
8. A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that
AB + CD = AD + BC
Fig. 10.12 Fig. 10.13
9. In Fig. 10.13, XY and XY are two parallel tangents to a circle with centre O and
another tangent AB with point of contact C intersecting XY at A and XY at B. Prove
that AOB = 90°.
10. Prove that the angle between the two tangents drawn from an external point to a circle
is supplementary to the angle subtended by the line-segment joining the points of
contact at the centre.
11. Prove that the parallelogram circumscribing a
circle is a rhombus.
12. A triangle ABC is drawn to circumscribe a circle
of radius 4 cm such that the segments BD and
DC into which BC is divided by the point of
contact D are of lengths 8 cm and 6 cm
respectively (see Fig. 10.14). Find the sides AB
and AC.
13. Prove that opposite sides of a quadrilateral
circumscribing a circle subtend supplementary
angles at the centre of the circle.
Fig. 10.14
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10.4 Summary
In this chapter, you have studied the following points :
1. The meaning of a tangent to a circle.
2. The tangent to a circle is perpendicular to the radius through the point of contact.
3. The lengths of the two tangents from an external point to a circle are equal.
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