Mathematics
Class-X
2015-2016
TANGENT PROPERTIES OF A CIRCLE
Theorem - 1 : The tangent is perpendicular to the radius through the point of contact.
Given : AB is a tangent at P to a circle with centre O and OP is a radius.
To Prove : OP is perpendicular to AB.
Construction : Any point Q is taken on the tangent AB and OQ is joined.
Proof :
Statement Reason
1. OP < OQ
2. OP is the shortest distance between O and
AB.
1. By definition of tangent any point on it other than P lie outside the circle.
2. Statement (1)
3. OP is perpendicular to AB 3. Perpendicular is the shortest distance from a point to a line.
Hence radius OP perpendicular to the tangent AB.
Theorem - 2 : If two circle touch each others, the point of contact lies on the straight line through the centres.
Given : Two circles with centres A and B touch each other externally in fig(i) and internally in fig (ii)
To Prove : P lies on the line joining A and B
Sudheer Gupta .
Be positive and constructive. 
Page 1

Mathematics
Class-X
Construction : A common tangent PT is drawn
Proof :
Statement
2015-2016
Reason
1. Tangent is perpendicular to the radius
1.
2. Reason (1)
2.
3. Statement (1) and (2)
3.
4. APB is a straight line. 4. Statement (3)
P lies on AB
Hence, P lies on AB.
Theorem - 3 : If a straight line touches a circle and from the point of contact a chord is drawn, the angle between the chord and the tangent are equal to the angles in the alternate segments.
Given : A line XY touches the circle at A and AB is a chord and O is the centre.
To Prove :
1.
point Q is taken in the alternate segment to the
Proof :
2.
Construction : A is joined with O and AO is produced to meet the circle at C. BC is joined. A and AQ, BQ are joined.
Statements Reason
(i)
1.
1. Tangent is perpendicular to the radius
2. Angle in a semi-circle
2.
Sudheer Gupta .
Be positive and constructive. 
Page 2

Mathematics
Class-X
2015-2016
3.
3. Sum of angles of a triangle.
4.
4. Statement (1) and (3)
5.
(ii)
5. Angles in the same segment are equal.
6. Opposite angles of a cyclic quadrilateral.
6.
7.
7. linear pair
8.
8. Statements (6) and (7)
9.
9. Statement (5)
Theorem - 4 : If a chord and a tangent intersect externally, then the product of the length of the segment of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
Given : The chord BA and the tangent PT at T intersect at P on t side the circle with centre O.
To Prove : PT
2
= PA . PB.
Construction : TA and TB are joined
Proof :
Statements
1.
2.
Sudheer Gupta .
Reason
1. (i) Angles in the alternate segment
(ii) Common
2. AA similarity
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Page 3

Mathematics
Class-X
3. Statements (2)
2015-2016
3.
Example - 1 : In the adjoining figure, PQ and PR are tangents to the circle, with centre O. If
, find i.
ii.
iii.
iv.
Solution : i.
OQ is perpendicular to PQ ii.
OR is perpendicular to PR
In quadrilateral PQOR,
Sudheer Gupta .
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Page 4

iii.
OQ = OR (radii)
Mathematics
Class-X
2015-2016 iv.
Example – 2 : If the sides of a quadrilateral PQRS touch a circle, prove that PQ + SR = QR + PS
Solution :
Given : Sides PQ, QR, RS and SP of a quadrilateral PQRS touch a circle at A, B, C and D respectively.
To Prove : PQ + SR = QR + PS.
Proof : Length of tangents are equal
PA = PD, QA = QB, RB = RC and SC = SD.
Now,
PQ + SR = PA + QA + SC + RC
= PD + QB + SD + RB
= (QB + RB) + (PD + SD)
= QR + PS.
Proved.
Sudheer Gupta .
Be positive and constructive. 
Page 5

Mathematics
Class-X
2015-2016
Sudheer Gupta .
Be positive and constructive. 
Page 6