Class 10 SELINA Solutions Maths Chapter 18 - Tangents and Intersecting Chords
Tangents and Intersecting Chords Exercise Ex. 18(A)
Solution 2
OP = 10 cm; radius OT = 8 cm
Length of tangent = 6 cm.
Solution 3
AB = 15 cm, AC = 7.5 cm
Let 'r' be the radius of the circle.
OC = OB = r
AO = AC + OC = 7.5 + r
In ∆AOB,
AO2 = AB2 + OB2
Therefore, the length of radius of a circle is 11.25 cm.
Solution 4
From Q, QA and QP are two tangents to the circle with centre O.
Therefore, QA = QP .....(i)
Similarly, from Q, QB and QP are two tangents to the circle with centre O'.
Therefore, QB = QP ......(ii)
From (i) and (ii)
QA = QB
Therefore, tangents QA and QB are equal.
Solution 5
Radius of outer circle, OS = 5 cm.
Radius of inner circle, OT = 3 cm.
(Angle between the radius and the tangent)
In Rt. triangle OTS, by Pythagoras Theorem,
Since OT is perpendicular to SP and OT bisects chord SP,
SP = 8 cm
Solution 6
AB = 6 cm, AC = 8 cm and BC = 9 cm
Let radii of the circles having centers A, B and C be r1, r2 and r3 respectively.
r1 + r3 = 8 ....(1)
r3 + r2 = 9 ....(2)
r2 + r1 = 6 ....(3)
Adding (1), (2) and (3),
r1 + r3 + r3 + r2 + r2 + r1 = 8 + 9 + 6
2(r1 + r2 + r3) = 23
r1 + r2 + r3 = 11.5 cm
r1 + 9 = 11.5 (Since r2 + r3 = 9)
r1 = 2.5 cm
r2 + 8 = 11.5 (Since r1 + r3 = 8)
r2 = 3.5 cm
r3 + 6 = 11.5 (Since r1 + r2 = 6)
r3 = 5.5 cm
Hence, r1 = 2.5 cm, r2 = 3.5 cm and r3 = 5.5 cm
Solution 7
Let the circle touch the sides AB, BC, CD and DA of quadrilateral ABCD at P, Q, R and S respectively.
Since AP and AS are tangents to the circle from external point A,
AP = AS ....(i)
Similarly, we can prove that:
BP = BQ ....(ii)
CR = CQ ....(iii)
DR = DS ....(iv)
Adding (i), (ii), (iii) and (iv),
AP + BP + CR + DR = AS + DS + BQ + CQ
AB + CD = AD + BC
Hence, AB + CD = AD + BC
Solution 8
From A, AP and AS are tangents to the circle.
Therefore, AP = AS ....(i)
Similarly, we can prove that
BP = BQ ....(ii)
CR = CQ ....(iii)
DR = DS ....(iv)
Adding,
AP + BP + CR + DR = AS + DS + BQ + CQ
AB + CD = AD + BC
But AB = CD and BC = AD ....(v) [Opposite sides of a parallelogram]
Therefore, AB + AB = BC + BC
2AB = 2BC
AB = BC ....(vi)
From (v) and (vi)
AB = BC = CD = DA
Hence, ABCD is a rhombus.
Solution 9
From B, BQ and BP are the tangents to the circle.
Therefore, BQ = BP ....(i)
Similarly, we can prove that
AP = AR ....(ii)
and CR = CQ ....(iii)
Adding,
AP + BQ + CR = BP + CQ + AR ....(iv) [Proved]
Adding (AP + BQ + CR) to both sides,
2(AP + BQ + CR) = AP + BP + CQ + BQ + AR + CR
2(AP + BQ + CR) = AB + BC + CA
Therefore, AP + BQ + CR = x (AB + BC + CA)
AP + BQ + CR = x Perimeter of triangle ABC
Solution 10
From A, AP and AR are the tangents to the circle.
Therefore, AP = AR.
Similarly, we can prove that
BP = BQ and CR = CQ
Adding,
AP + BP + CQ = AR + BQ + CR
(AP + BP) + CQ = (AR + CR) + BQ
AB + CQ = AC + BQ
But AB = AC
Therefore, BQ = CQ
Solution 11
Radius of bigger circle = 6.3 cm
Radius of smaller circle = 3.6 cm
i)
Two circles are touching each other at P externally.
O and O’ are the centers of the circles.
Join OP and O’P
OP = 6.3 cm, O’P = 3.6 cm
Distance between the centres = OO'
= OP + O’P
= 6.3 + 3.6
= 9.9 cm
ii)
Two circles are touching each other at P internally.
O and O’ are the centers of the circles.
Join OP and O’P.
OP = 6.3 cm, O’P = 3.6 cm
Distance between the centres = OO’
= OP - O’P
= 6.3 - 3.6
= 2.7 cm
Solution 12
i) In
AP = BP (Tangents from P to the circle)
OP = OP (Common)
OA = OB (Radii of the same circle)
ii) In
OA = OB (Radii of the same circle)
(Proved )
OM = OM (Common)
Hence, OM or OP is the perpendicular bisector of chord AB.
Solution 13
Draw TPT' as common tangent to the circles.
i) TA and TP are the tangents to the circle with centre O.
Therefore, TA = TP ....(i)
Similarly, TP = TB ....(ii)
From (i) and (ii)
TA = TB
Therefore, TPT' is the bisector of AB.
ii) Now in
Similarly in
Adding,
Solution 14
In quadrilateral OPAQ,
Now,
In triangle OPQ,
OP = OQ (Radii of the same circle)
From (i) and (ii)
Solution 15
In
Therefore, LBMO is a square.
LB = BM = OM = OL = x
Since ABC is a right triangle
Solution 16
The incircle touches the sides of the triangle ABC and
i) In quadrilateral AROQ,
ii) Now arc RQ subtends at the centre and at the remaining part of the circle.
Solution 17
Join QR.
i) In quadrilateral ORPQ,
ii) In
OQ = QR (Radii of the same circle)
iii) Now arc RQ subtends at the centre and at the remaining part of the circle.
Solution 18
In
OB = OC (Radii of the same circle)
Now in
Solution 19
(angles in alternate segment)
But OS = OR (Radii of the same circle)
Now,
OQ = OR (radii of same circle)
But in
From (i) and (ii),
Solution 20
Join AT and BT.
i) TC is the diameter of the circle
(Angle in a semi-circle)
ii)
(Angles in the same segment of the circle)
(Angles in the same segment of the circle)
iii) (angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment)
Now in
Solution 21
Join OC.
Now, PA and PC are the tangents to the circle with centre O.
In quadrilateral APCO,
Now, arc BC subtends at the centre and at the remaining part of the circle
Solution 22
∠CAB = ∠BAQ = 30° (AB is angle bisector of ∠CAQ)
Then, ∠CAQ = 2∠BAQ = 60°
Now, ∠CAQ + ∠PAC = 180° (angles in linear pair)
∴ ∠PAC = 180° - 60° = 120°
∠PAC = 2∠CAD (AD is angle bisector of ∠PAC)
∠CAD = 60°
Now,
∠DAB = ∠CAD + ∠CAB = 60° + 30° = 90°
Thus, BD subtends a right angle on the circle.
So, BD is the diameter of the circle.
Tangents and Intersecting Chords Exercise Ex. 18(B)
Solution 2
Solution 3
i) PAQ is a tangent and AB is the chord.
(angles in the alternate segment)
ii) OA = OD (radii of the same circle)
iii) BD is the diameter.
(angle in a semi-circle)
Now in
Solution 4
PQ is a tangent and OR is the radius.
But in
OT = OR (Radii of the same circle)
In
Solution 5
Join O'A and O'B.
CD is the tangent and AO is the chord.
(angles in alternate segment)
In
OA = OB (Radii of the same circle)
From (i) and (ii)
Therefore, OA is bisector of BAC.
Solution 6
Join OC, OD and OA.
i)
ii)
PCT is a tangent and CA is a chord.
But arc DC subtends at the centre and at the
remaining part of the circle.
Solution 7
i) PA is the tangent and AB is a chord
( angles in the alternate segment)
AD is the bisector of
In
Therefore, is an isosceles triangle.
ii) In
Solution 8
Join AB.
(angles in alternate segment)
Similarly,
Adding (i) and (ii),
From (iii) and (iv)
Hence, and are supplementary.
Solution 9
Join AB.
i) In Rt.
Chords AE and CB intersect each other at D inside the circle.
Therefore,
AD x DE = BD x DC
3 x DE = 4 x 9
DE = 12 cm
ii) Given, AD = BD ....(i)
We know that:
AD x DE = BD x DC
But AD = BD
Therefore, DE = DC ....(ii)
Adding (i) and (ii)
AD + DE = BD + DC
Therefore, AE = BC
Solution 10
AB is a tangent and hence, is a straight line.
AB i.e. DB is tangent to the circle at point B and BC is the diameter.
Now, OE = OC (radii of the same circle)
(vertically opposite angles)
In
Tangents and Intersecting Chords Exercise TEST YOURSELF
Solution 2
Given: A circle with centre O and radius r. . Also AB > CD
To prove: OM < ON
Proof: Join OA and OC.
In Rt.
Again in Rt.
From (i) and (ii)
Hence, AB is nearer to the centre than CD.
Solution 3
ABCD is a cyclic quadrilateral in which AD||BC
(Sum of opposite angles of a quadrilateral)
Now in
Now in
Solution 4
Since ABCD is a cyclic quadrilateral, therefore, BCD + BAD = 180
(since opposite angles of a cyclic quadrilateral are supplementary)
BCD + 70 = 180
BCD = 180 - 70 = 110
In BCD, we have,
CBD + BCD + BDC = 180
30 + 110 + BDC = 180
BDC = 180 - 140
BDC = 40
Solution 5
ABCD is a cyclic quadrilateral.
Similarly,
Hence,
Solution 6
Join AD.
AB is the diameter.
ADB = 90º (Angle in a semi-circle)
But, ADB + ADC = 180º (linear pair)
ADC = 90º
In ABD and ACD,
ADB = ADC (each 90º)
AB = AC (Given)
AD = AD (Common)
ABD ACD (RHS congruence criterion)
BD = DC (C.P.C.T)
Hence, the circle bisects base BC at D.
Solution 7
Join ED, EF and DF. Also join BF, FA, AE and EC.
In cyclic quadrilateral AFBE,
(Sum of opposite angles)
Similarly in cyclic quadrilateral CEAF,
Adding (ii) and (iii)
Solution 8
Join OB.
BC = OD ... (given)
OD = OB ... (radii of the same circle)
Hence,
BC = OB
In
In
Since DOC is a straight-line
Solution 9
Join AP and BP.
Since TPS is a tangent and PA is the chord of the circle.
(angles in alternate segments)
But
arc PA = arc PB,
hence PA = PB
But these are alternate angles
Solution 10
i) PQ is tangent and CD is a chord
(angles in the alternate segment)
ii)
iii) In
Solution 11
Join OC.
BCD is the tangent and OC is the radius.
Substituting in (i)
Solution 12
i) In
and BC is the diameter of the circle.
Therefore, AB is the tangent to the circle at B.
Now, AB is tangent and ADC is the secant
Now,
∠BDC = ∠BDA = 90o ... angle subtended by a diameter
ii) In
From (i) and (ii)
Now in
Solution 13
In
(angles in the same segment)
AC = AE (Given)
(Common)
(ASA Postulate)
AB = AD
but AC = AE
In
(angles in the same segment)
BC = DE
(angles in the same segment)
(ASA Postulate)
BP = DP and CP = PE (cpct)
Solution 14
i) Join OC and OB.
AB = BC = CD and
OA = OD = OC = OB all are radii of the circle
Hence,
All the 3 triangles are congruent
So,
∠AOB = ∠DOC = ∠COB [By C.P.C.T.C]
OB and OC are the bisectors of and respectively.
In
Hence,
∠AOB = ∠DOC = ∠COB = 60o
Arc BC subtends at the centre and at the remaining part of the circle.
ii)
Similarly,
∠BED = 0.5 ∠BOD = 60o
Solution 15
In the given fig, O is the centre of the circle and CA and CB are the tangents to the circle from C. Also, ACO = 30
P is any point on the circle. P and PB are joined.
To find: (i)
(ii)
(iii)
Proof:
Solution 16
Join PB.
i) In cyclic quadrilateral PBCQ,
Now in
In cyclic quadrilateral PQBA,
ii) Now in
iii) Arc AQ subtends at the centre and APQ at the remaining part of the circle.
We have,
From (1), (2) and (3), we have
Now in
But these are alternate angles.
Hence, AO is parallel to BQ.
Solution 17
Join PQ, RQ and ST.
i)
Arc RQ subtends at the centre and QTR at the remaining part of the circle.
ii) Arc QP subtends at the centre and QRP at the remaining part of the circle.
iii) RS || QT
iv) Since RSTQ is a cyclic quadrilateral
(sum of opposite angles)
Solution 18
Join PB.
In TAP and TBP,
TA = TB (tangents segments from an external points are equal in length)
Also, ATP = BTP. (since OT is equally inclined with TA and TB) TP = TP (common)
TAP TBP (by SAS criterion of congruency)
TAP = TBP (corresponding parts of congruent triangles are equal)
But TBP = BAP (angles in alternate segments)
Therefore, TAP = BAP.
Hence, AP bisects TAB.
Solution 19
We know that XB.XA = XD.XC
Or, XB.(XB + BA) = XD.(XD + CD)
Or, 6(6 + 4) = 5(5 + CD)
Or, 60 = 5(5 + CD)
Or, 5 + CD = = 12
Or, CD = 12 - 5 = 7 cm.
Solution 20
From the figure we see that BQ = BR = 27 cm (since length of the tangent segments from an
external point are equal)
As BC = 38 cm
CR = CB - BR = 38 - 27
= 11 cm
Again,
CR = CS = 11cm (length of tangent segments from an external point are equal)
Now, as DC = 25 cm
DS = DC - SC
= 25 -11
= 14 cm
Now, in quadrilateral DSOP,
PDS = 90 (given)
OSD = 90, OPD = 90 (since tangent is perpendicular to the
radius through the point of contact)
DSOP is a parallelogram
OP||SD and PD||OS
Now, as OP = OS (radii of the same circle)
OPDS is a square. DS = OP = 14cm
radius of the circle = 14 cm
Solution 21
In AXB,
XAB + AXB + ABX=180 [Triangle property]
XAB + 50 + 70 = 180
XAB=180 - 120 = 60
XAY=90 [Angle of semi-circle]
BAY=XAY -XAB = 90 - 60 = 30
and BXY = BAY = 30 [Angle of same segment]
ACX = BXY + ABX [External angle = Sum of two interior angles]
= 30 + 70
= 100
also,
XYP=90 [Diameter ⊥ tangent]
APY = ACX -CYP
APY=100 - 90
APY=10
Solution 22
PAQ is a tangent and AB is a chord of the circle.
i) (angles in alternate segment)
ii) In
iii) (angles in the same segment)
Now in
iv) PAQ is the tangent and AD is chord
Solution 23
i) AB is the diameter of the circle.
In
ii) QC is tangent to the circle
The angle between tangent and chord = angle in the alternate segment
ABQ is a straight-line
In ΔCBQ
Solution 24
i)
ii) Since, BPDO is a quadrilateral,
Solution 25
i) PQ = RQ
(opposite angles of equal sides of a triangle)
Now, QOP = 2PRQ (angle at the centre is double)
ii) PQC = PRQ (angles in alternate segments are equal)
QPC = PRQ (angles in alternate segments)
Solution 26
Since AC is tangent to the circle with center P at point A.
In
Also in Rt.
From (i) and (ii),
Solution 27
In the figure, a circle with centre O, is the circumcircle of triangle XYZ.
Tangents at X and Y intersect at point T, such that XTY = 80o
Since a tangent at any point of a circle is perpendicular to the radius at the point of contact,
we have ∠OXT = ∠OYT = 90o
So in quadrilateral XOYT
Sum of angles of a quadrilateral is 360 degrees,
∠XOY = 360 − (90 + 90 + 80)
Solution 28
From Rt.
Now, since the two chords AE and BC intersect at D,
AD x DE = CD x DB
3 x DE = 9 x 4
Hence, AE = AD + DE = (3 + 12) = 15 cm
Solution 29
Solution 30