As the chapter name suggests, this chapter is about tangents and intersecting chords. Important theorems related to tangents and intersecting chords are discussed under this chapter. Students facing any difficulties in understanding concepts of this chapter and others can refer to the Selina Solutions for Class 10 Mathematics developed by subject matter experts at BYJUâ€™S. All the solutions are created with an aim to boost confidence for their ICSE preparations. The resource also improves problem-solving skills of the students, which are essential from an examination standpoint. The Selina Solutions for Class 10 Mathematics Chapter 18 Tangents and Intersecting Chords PDF can be accessed from the link given below.

## Selina Solutions Concise Maths Class 10 Chapter 18 Tangents and Intersecting Chords Download PDF

### Exercises of Concise Selina Solutions Class 10 Maths Chapter 18 Tangents and Intersecting Chords

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Exercise 18(A) Page No: 274

**1. The radius of a circle is 8cm. Calculate the length of a tangent drawn to this circle from a point at a distance of 10cm from its centre. **

**Solution: **

Given, a circle with centre O and radius 8 cm.

An external point P from where a tangent is drawn to meet the circle at T.

OP = 10 cm; radius OT = 8 cm

As OT âŠ¥ PT

In right âˆ†OTP, we have

OP^{2} = OT^{2} + PT^{2} [By Pythagoras Theorem]

10^{2} = 8^{2} + PT^{2}Â

PT^{2} = 100 â€“ 64 = 36

So, PT = 6

Therefore, length of tangent = 6 cm.

**2. In the given figure, O is the centre of the circle and AB is a tangent to the circle at B. If AB = 15 cm and AC = 7.5 cm, calculate the radius of the circle.**

**Solution: **

Given,

AB = 15 cm, AC = 7.5 cm

Letâ€™s assume the radius of the circle to be ‘r’.

So, AO = AC + OC = 7.5 + r

In right âˆ†AOB, we have

AO^{2}Â = AB^{2}Â + OB^{2 } [By Pythagoras Theorem]

(7.5 + r)^{2 }= 15^{2} + r^{2}

56.25 + r^{2} + 15r = 225 + r^{2}

15r = 225 â€“ 56.25

r = 168.75/ 15

Thus,

r = 11.25 cm

**3. Two circles touch each other externally at point P. Q is a point on the common tangent through P. Prove that the tangents QA and QB are equal.**

**Solution: **

Let Q be the point from which, QA and QP are two tangents to the circle with centre O

So, QA = QP …..(a)

Similarly, from point Q, QB and QP are two tangents to the circle with centre O’

So, QB = QP ……(b)

From (a) and (b), we have

QA = QB

Therefore, tangents QA and QB are equal.

– Hence Proved

**4. Two circles touch each other internally. Show that the tangents drawn to the two circles from any point on the common tangent are equal in length.**

**Solution:**

Let Q be the point on the common tangent from which, two tangents QA and QP are drawn to the circle with centre O.

So, QA = QP ……. (1)

Similarly, from point Q, QB and QP are two tangents to the circle with centre O’

So, QB = QP ……. (2)

From (1) and (2), we have

QA = QB

Therefore, tangents QA and QB are equal.

– Hence Proved

**5. Two circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner.**

**Solution:**

Given,

OS = 5 cm and OT = 3 cm

In right triangle OST, we have

ST^{2} = OS^{2} â€“ OT^{2}

= 25 â€“ 9

= 16

So, ST = 4 cm

As we know, OT is perpendicular to SP and OT bisects chord SP

Hence, SP = 2 x ST = 8 cm

**6. Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.**

**Solution: **

Let ABC be the triangle formed when centres of 3 circles are joined.

Given,

AB = 6 cm, AC = 8 cm and BC = 9 cm

And let the radii of the circles having centres A, B and C be r_{1}, r_{2}Â and r_{3}Â respectively.

So, we have

r_{1}Â + r_{3}Â = 8

r_{3}Â + r_{2}Â = 9

r_{2}Â + r_{1}Â = 6

Adding all the above equations, we get

r_{1}Â + r_{3}Â + r_{3}Â + r_{2}Â + r_{2}Â + r_{1}Â = 8 + 9 + 6

2(r_{1}Â + r_{2}Â + r_{3}) = 23

So,

r_{1}Â + r_{2}Â + r_{3Â }= 11.5 cm

Now,

r_{1}Â + 9 = 11.5 (As r_{2}Â + r_{3Â }= 9)

r_{1}Â = 2.5 cm

And,

r_{2}Â + 6 = 11.5 (As r_{1}Â + r_{3Â }= 6)

r_{2}Â = 5.5 cm

Lastly, r_{3}Â + 8 = 11.5 (As r_{2}Â + r_{1Â }= 8)

r_{3Â }= 3.5 cm

Therefore, the radii of the circles are r_{1}Â = 2.5 cm, r_{2}Â = 5.5 cm and r_{3Â }= 3.5 cm.

**7. If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.**

**Solution:**

Let a circle touch the sides AB, BC, CD and DA of quadrilateral ABCD at P, Q, R and S respectively.

As, AP and AS are tangents to the circle from an external point A, we have

AP = AS ……. (1)

Similarly, we also get

BP = BQ ……. (2)

CR = CQ ……. (3)

DR = DS …….. (4)

Adding (1), (2), (3) and (4), we get

AP + BP + CR + DR = AS + DS + BQ + CQ

AB + CD = AD + BC

Therefore,

AB + CD = AD + BC

– Hence Proved

**8. If the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.**

**Solution:**

Let a circle touch the sides AB, BC, CD and DA of parallelogram ABCD at P, Q, R and S respectively.

Now, from point A, AP and AS are tangents to the circle.

So, AP = AS……. (1)

Similarly, we also have

BP = BQ ……… (2)

CR = CQ ……… (3)

DR = DS ……… (4)

Adding (1), (2), (3) and (4), we get

AP + BP + CR + DR = AS + DS + BQ + CQ

AB + CD = AD + BC

Therefore,

AB + CD = AD + BC

But AB = CD and BC = AD……. (5) [Opposite sides of a parallelogram]

Hence,

AB + AB = BC + BC

2AB = 2 BC

AB = BC …….. (6)

From (5) and (6), we conclude that

AB = BC = CD = DA

Thus, ABCD is a rhombus.

**9. From the given figure prove that:**

**AP + BQ + CR = BP + CQ + AR.**

**Also, show that AP + BQ + CR =Â Â½ x perimeter of triangle ABC.**

**Solution: **

As from point B, BQ and BP are the tangents to the circle

We have, BQ = BP â€¦â€¦â€¦..(1)

Similarly, we also get

AP = AR â€¦â€¦â€¦â€¦.. (2)

And, CR = CQ â€¦â€¦ (3)

Adding (1), (2) and (3) we get,

AP + BQ + CR = BP + CQ + AR â€¦â€¦â€¦ (4)

Now, adding AP + BQ + CR to both sides in (4), we get

2(AP + BQ + CR) = AP + PQ + CQ + QB + AR + CR

2(AP + BQ + CR) = AB + BC + CA

Therefore, we get

AP + BQ + CR =Â Â½ x (AB + BC + CA)

i.e.

AP + BQ + CR =Â Â½ x perimeter of triangle ABC

**10. In the figure, if AB = AC then prove that BQ = CQ.**

**Solution:**

As, from point A

AP and AR are the tangents to the circle

So, we have AP = AR

Similarly, we also have

BP = BQ and CR = CQ [From points B and C]

Now adding the above equations, we get

AP + BP + CQ = AR + BQ + CR

(AP + BP) + CQ = (AR + CR) + BQ

AB + CQ = AC + BQ â€¦.. (i)

But, as AB = AC [Given]

Therefore, from (i)

CQ = BQ or BQ = CQ

**11. Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if –**

**i) they touch each other externally.**

**ii) they touch each other internally.**

**Solution: **

Given,

Radius of bigger circle = 6.3 cm and of smaller circle = 3.6 cm

i)

When the two circles touch each other at P externally. O and Oâ€™ are the centers of the circles. Join OP and Oâ€™P.

So, OP = 6.3 cm, Oâ€™P = 3.6 cm

Hence, the distance between their centres (OOâ€™) is given by

OOâ€™ = OP + Oâ€™P = 6.3 + 3.6 = 9.9 cm

ii)

When the two circles touch each other at P internally. O and Oâ€™ are the centers of the circles. Join OP and Oâ€™P

So, OP = 6.3 cm, Oâ€™P = 3.6 cm

Hence, the distance between their centres (OOâ€™) is given by

OOâ€™ = OP – Oâ€™P = 6.3 – 3.6 = 2.7 cm

**12. From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:**

**i) **âˆ AOP = âˆ BOP**Â **

**ii) OP is the âŠ¥ bisector of chord AB.**

**Solution: **

i) InÂ âˆ†AOP and âˆ†BOP, we have

AP = BP [Tangents from P to the circle]

OP = OP [Common]

OA = OB [Radii of the same circle]

Hence, by SAS criterion of congruence

Î”AOP â‰… Î”BOP

So, by C.P.C.T we have

âˆ AOP = âˆ BOP

ii) In âˆ†OAM and âˆ†OBM, we have

OA = OB [Radii of the same circle]

âˆ AOM = âˆ BOM Â [ProvedÂ âˆ AOP = âˆ BOP]

OM = OM [Common]

Hence, by SAS criterion of congruence

Î”OAM â‰… Î”OBM

So, by C.P.C.T we have

AM = MB

And âˆ OMA = âˆ OMB

But,

âˆ OMA + âˆ OMB = 180^{o}

Thus, âˆ OMA = âˆ OMB = 90^{o}

Therefore, OM or OP is the perpendicular bisector of chord AB.

– Hence Proved

Exercise 18(B) Page No: 283

**1. (i) In the given figure, 3 x CP = PD = 9 cm and AP = 4.5 cm. Find BP.**

** **

**(ii) In the given figure, 5 x PA = 3 x AB = 30 cm and PC = 4cm. Find CD.**

**(iii) In the given figure, tangent PT = 12.5 cm and PA = 10 cm; find AB.**

**Solution: **

(i) As the two chords AB and CD intersect each other at P, we have

AP x PB = CP x PD

4.5 x PB = 3 x 9 [3CP = 9 cm so, CP = 3 cm]

PB = (3 x 9)/ 4.5 = 6 cm

(ii) As the two chords AB and CD intersect each other at P, we have

AP x PB = CP x PD

But, 5 x PA = 3 x AB = 30 cm

So, PA = 30/5 = 6 cm and AB = 30/3 = 10 cm

And, BP = PA + AB = 6 + 10 = 16 cm

Now, as

AP x PB = CP x PD

6 x 16 = 4 x PD

PD = (6 x 16)/ 4 = 24 cm

CD = PD â€“ PC = 24 â€“ 4 = 20 cm

(iii) As PAB is the secant and PT is the tangent, we have

PT^{2} = PA x PB

12.5^{2} = 10 x PB

PB = (12.5 x 12.5)/ 10 = 15.625 cm

AB = PB â€“ PA = 15.625 â€“ 10 = 5.625 cm

**2. In the given figure, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. **

**Find**

**(i) AB.**

**(ii)Â theÂ length of tangent PT.**

**Solution: **

(i) PA = AB + BP = (AB + 4) cm

PC = PD + CD = 5 + 7.8 = 12.8 cm

As PA x PB = PC x PD

(AB + 4) x 4 = 12.8 x 5

AB + 4 = (12.8 x 5)/ 4

AB + 4 = 16

Hence, AB = 12 cm

(ii) As we know,

PT^{2} = PC x PD

PT^{2} = 12.8 x 5 = 64

Thus, PT = 8 cm

**3. In the following figure, PQ is the tangent to the circle at A, DB is a diameter and O is the centre of the circle. IfÂ âˆ ADB = 30 ^{o} and âˆ CBD = 60^{o}; calculate:**

**i) âˆ QABÂ **

**ii)Â âˆ PAD **

**iii)Â âˆ CDB **

**Solution: **

(i) Given, PAQ is a tangent and AB is the chord

âˆ QABÂ = âˆ ADB = 30^{o} [Angles in the alternate segment]

(ii) OA = OD [radii of the same circle]

So, âˆ OAD = âˆ ODA = 30^{o}

But, as OA âŠ¥ PQ

âˆ PAD = âˆ OAP – âˆ OAD = 90^{o} â€“ 30^{o} = 60^{o}

(iii) As BD is the diameter, we have

âˆ BCD = 90^{o} [Angle in a semi-circle]

Now in âˆ†BCD,

âˆ CDB + âˆ CBD + âˆ BCD = 180^{o}

âˆ CDB + 60^{o} + 90^{o} = 180^{o}

Thus, âˆ CDB = 180^{o} – 150^{o} = 30^{o}

**4. If PQ is a tangent to the circle at R; calculate:**

**i)Â âˆ PRS**

**ii)Â âˆ ROT**

**Given: O is the centre of the circle and **âˆ TRQ = 30^{o}

**Solution: **

(i) As PQ is the tangent and OR is the radius.

So, OR âŠ¥ PQ

âˆ ORT = 90^{o}

âˆ TRQ = 90^{o} â€“ 30^{o} = 60^{o}

But in âˆ†OTR, we have

OT = OR [Radii of same circle]

âˆ OTR = 60^{o} or âˆ STR = 60^{o}

But,

âˆ PRS = âˆ STR = 60^{o} [Angles in the alternate segment]

(ii) In âˆ†OTR,

âˆ ORT = 60^{o}Â

âˆ OTR = 60^{o}

Thus,

âˆ ROT = 180^{o} â€“ (60^{o} + 60^{o}) = 180^{o} â€“ 120^{o} = 60^{o}

**5. AB is diameter and AC is a chord of a circle with centre O such that angle BAC=30Âº. The tangent to the circle at C intersects AB produced in D. Show that BC = BD.**

**Solution: **

Join OC.

âˆ BCD = âˆ BAC = 30^{o} [Angles in the alternate segment]

Itâ€™s seen that, arc BC subtends âˆ DOC at the center of the circle and âˆ BAC at the remaining part of the circle.

So, âˆ BOC = 2âˆ BAC = 2 x 30^{o} = 60^{o}

Now, in âˆ†OCD

âˆ BOC or âˆ DOC = 60^{o}

âˆ OCD = 90^{o} [OC âŠ¥ CD]

âˆ DOC + âˆ ODC = 90^{o}

âˆ ODC = 90^{o} â€“ 60^{o} = 30^{o}

Now, in âˆ†BCD

As âˆ ODC or âˆ BDC = âˆ BCD = 30^{o}

Therefore, BC = BD

**6. Tangent at P to the circumcircle of triangle PQR is drawn. If this tangent is parallel to side QR, show that triangle PQR is isosceles.**

**Solution: **

Let DE be the tangent to the circle at P.

And, DE || QR [Given]

âˆ EPR = âˆ PRQ [Alternate angles are equal]

âˆ DPQ = âˆ PQR [Alternate angles are equal] â€¦.. (i)

Let âˆ DPQ = x and âˆ EPR = y

As the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment, we have

âˆ DPQ = âˆ PRQ â€¦â€¦ (ii) [DE is tangent and PQ is chord]

So, from (i) and (ii),

âˆ PQR = âˆ PRQ

PQ = PR

Therefore, triangle PQR is an isosceles triangle.

**7. Two circles with centres O and O’ are drawn to intersect each other at points A and B.**

**Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O’ at A. Prove that OA bisects angle BAC.**

**Solution: **

Join OA, OB, O’A, O’B and O’O.

CD is the tangent and AO is the chord.

âˆ OAC = âˆ OBA â€¦ (i)Â [Angles in alternate segment]

InÂ âˆ†OAB,

OA = OB [Radii of the same circle]

âˆ OAB = âˆ OBA â€¦.. (ii)

From (i) and (ii), we have

âˆ OAC = âˆ OAB

Thus, OA is the bisector ofÂ âˆ BAC.

**8. Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Prove that:Â âˆ CPA = âˆ DPB**

**Solution: **

Letâ€™s draw a tangent TS at P to the circles given.

As TPS is the tangent and PD is the chord, we have

âˆ PAB = âˆ BPS â€¦. (i) [Angles in alternate segment]

Similarly,

âˆ PCD = âˆ DPS â€¦. (ii)

Now, subtracting (i) from (ii) we have

âˆ PCD – âˆ PAB = âˆ DPS – âˆ BPS

But inÂ âˆ†PAC,

Ext. âˆ PCD = âˆ PAB + âˆ CPA

âˆ PAB + âˆ CPA – âˆ PAB = âˆ DPS – âˆ BPS

Thus,

âˆ CPA = âˆ DPB

Exercise 18(C) Page No: 285

**1. Prove that, of any two chords of a circle, the greater chord is nearer to the center.**

**Solution: **

Given: A circle with center O and radius r. AB and CD are two chords such that AB > CD. Also, OM âŠ¥ AB and ON âŠ¥ CD.

Required to prove: OM < ON

Proof:

Join OA and OC.

Then in right âˆ†AOM, we have

AO^{2}Â = AM^{2} + OM^{2}

r^{2} = (Â½AB)^{2} + OM^{2}

r^{2} = Â¼ AB^{2} + OM^{2} â€¦.. (i)

Again, in right âˆ†ONC, we have

OC^{2}Â = NC^{2} + ON^{2}

r^{2} = (Â½CD)^{2} + ON^{2}

r^{2} = Â¼ CD^{2} + ON^{2} â€¦.. (ii)

On equating (i) and (ii), we get

Â¼ AB^{2} + OM^{2} = Â¼ CD^{2} + ON^{2}

But, AB > CD [Given]

So, ON will be greater than OM to be equal on both sides.

Thus,

OM < ON

Hence, AB is nearer to the centre than CD.

**2. OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O.**

**i) If the radius of the circle is 10 cm, find the area of the rhombus.**

**ii) If the area of the rhombus is 32âˆš3 cm ^{2}, find the radius of the circle.**

**Solution: **

(i) Given, radius = 10 cm

In rhombus OABC,

OC = 10 cm

So,

OE = Â½ x OB = Â½ x 10 = 5 cm

Now, in right âˆ†OCE

OC^{2} = OE^{2}Â + EC^{2}

10^{2} = 5^{2}Â + EC^{2}

EC^{2} = 100 â€“ 25 = 75

EC = âˆš75 = 5âˆš3

Hence, AC = 2 x EC = 2 x 5âˆš3 = 10âˆš3

We know that,

Area of rhombus = Â½ x OB x AC

= Â½ x 10 x 10âˆš3

= 50âˆš3 cm^{2} â‰ˆ 86.6 cm^{2}

(ii) We have the area of rhombus = 32âˆš3 cm^{2}

But area of rhombus OABC = 2 x area of âˆ†OAB

Area of rhombus OABC = 2 x (âˆš3/4) r^{2}

Where r is the side of the equilateral triangle OAB.

2 x (âˆš3/4) r^{2} = 32âˆš3

âˆš3/2 r^{2} = 32âˆš3

r^{2} = 64

r = 8

Therefore, the radius of the circle is 8 cm.

**3. Two circles with centers A and B, and radii 5 cm and 3 cm, touch each other internally. If the perpendicular bisector of the segment AB meets the bigger circle in P and Q; find the length of PQ.**

**Solution: **

We know that,

If two circles touch internally, then distance between their centres is equal to the difference of their radii. So, AB = (5 – 3) cm = 2 cm.

Also, the common chord PQ is the perpendicular bisector of AB.

Thus, AC = CB = Â½ AB = 1 cm

In rightÂ âˆ†ACP, we have

AP^{2}Â = AC^{2}Â + CP^{2} [Pythagoras Theorem]

5^{2}Â = 1^{2}Â + CP^{2}

CP^{2}Â = 25 – 1 = 24

CP = âˆš24 cm = 2âˆš6 cm

Now,

PQ = 2 CP

= 2 xÂ 2âˆš6 cm

= 4âˆš6 cm

Therefore, the length of PQ is 4âˆš6 cm.

**4. Two chords AB and AC of a circle are equal. Prove that the center of the circle, lies on the bisector of the angle BAC.**

**Solution:**

Given:Â AB and AC are two equal chords of C (O, r).

Required to prove:Â Centre, O lies on the bisector ofÂ âˆ BAC.

Construction:Â Join BC. Let the bisector ofÂ âˆ BAC intersects BC in P.

Proof:

InÂ âˆ†APB andÂ âˆ†APC,

AB = AC [Given]

âˆ BAP =Â âˆ CAP [Given]

AP = AP [Common]

Hence, âˆ†APB â‰… âˆ†APC by SAA congruence criterion

So, by CPCT we have

BP = CP andÂ âˆ APB =Â âˆ APC

And,

âˆ APB +Â âˆ APC = 180

Â [Linear pair]

2âˆ APB = 180

Â [âˆ APB =Â âˆ APC]

âˆ APB = 90

Now, BP = CP andÂ âˆ APB = 90

Therefore, AP is the perpendicular bisector of chord BC.

Hence, AP passes through the centre, O of the circle.

**5. The diameter and a chord of circle have a common end-point. If the length of the diameter is 20 cm and the length of the chord is 12 cm, how far is the chord from the center of the circle?**

**Solution: **

We have, AB as the diameter and AC as the chord.

Now, drawÂ OL âŠ¥ AC

Since OL âŠ¥ ACÂ and hence it bisects AC, O is the centre of the circle.

Therefore, OA = 10 cm and AL = 6 cm

Now, in right âˆ†OLA

AO^{2}Â = AL^{2Â }+ OL^{2} [By Pythagoras Theorem]

10^{2} = 6^{2} + OL^{2}

OL^{2} = 100 â€“ 36 = 64

OL = 8 cm

Therefore, the chord is at a distance of 8 cm from the centre of the circle.

**6. ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110 ^{o}Â and angle BAC =Â 50^{o}. Find angle DAC and angle DCA.**

**Solution: **

Given, ABCD is a cyclic quadrilateral in which AD || BC

And, âˆ ADC = 110^{o}, âˆ BAC = 50^{o}

We know that,

âˆ B + âˆ D = 180^{o} [Sum of opposite angles of a quadrilateral]

âˆ B + 110^{o} = 180^{o}

So, âˆ B = 70^{o}

Now in âˆ†ADC, we have

âˆ BAC + âˆ ABC + âˆ ACB = 180^{o}

50^{o} + 70^{o} + âˆ ACB = 180^{o}

âˆ ACB = 180^{o} â€“ 120^{o} = 60^{o}

And, as AD || BC we have

âˆ DAC = âˆ ACB = 60^{o} [Alternate angles]

Now in âˆ†ADC,

âˆ DAC + âˆ ADC + âˆ DCA = 180^{o}

60^{o} + 110^{o} + âˆ DCA = 180^{o}

Thus,

âˆ DCA = 180^{o} â€“ 170^{o} = 10^{o}Â

**7. In the given figure, C and D are points on the semi-circle described on AB as diameter.**

**Given angle BAD = 70 ^{o} and angle DBC = 30^{o}, calculate angle BDC.**

**Solution: **

As ABCD is a cyclic quadrilateral, we have

âˆ BCD +Â âˆ BAD = 180

[Opposite angles of a cyclic quadrilateral are supplementary]

âˆ BCD + 70^{o}Â = 180

âˆ BCD = 180^{o}Â â€“ 70^{o}Â = 110^{o}

And, by angle sum property of âˆ†BCD we have

âˆ CBD +Â âˆ BCD +Â âˆ BDC = 180^{o}

30^{o}Â + 110^{o}Â +Â âˆ BDC = 180^{o}

âˆ BDC = 180^{o }– 140^{o}

Thus,

âˆ BDC = 40^{o}

**8. In cyclic quadrilateral ABCD,Â **âˆ **A = 3 **âˆ **C andÂ **âˆ **D = 5 **âˆ **B. Find the measure of each angle of the quadrilateral.**

**Solution: **

Given, cyclic quadrilateral ABCD

So, âˆ A + âˆ C = 180^{o} [Opposite angles in a cyclic quadrilateral is supplementary]

3âˆ C + âˆ C = 180^{o} [As âˆ A = 3 âˆ C]

âˆ C = 45^{o}

Now,

âˆ A = 3 âˆ C = 3 x 45^{o}

âˆ A = 135^{o}Â

Similarly,

âˆ B + âˆ D = 180^{oÂ }[As âˆ D = 5 âˆ B]

âˆ B + 5âˆ B = 180^{o}

6âˆ B = 180^{o}

âˆ B = 30^{o}

Now,

âˆ D = 5âˆ B = 5 x 30^{o}

âˆ D = 150^{o}

Therefore,

âˆ A = 135^{o}, âˆ B = 30^{o}, âˆ C = 45^{o}, âˆ D = 150^{o}Â

**9. Show that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.**

**Solution: **

Letâ€™s join AD.

And. AB is the diameter.

We have âˆ ADB = 90ÂºÂ [Angle in a semi-circle]

But,Â

âˆ ADB +Â âˆ ADC = 180ÂºÂ [Linear pair]

So, âˆ ADC = 90Âº

Now, inÂ âˆ†ABD andÂ âˆ†ACD we have

âˆ ADB =Â âˆ ADC [each 90Âº]

AB = AC [Given]

AD = AD [Common]

Hence, âˆ†ABDÂ â‰…Â âˆ†ACDÂ by RHS congruence criterion

So, by C.P.C.T

BD = DC

Therefore, the circle bisects base BC at D.

**10. Bisectors of vertex angles A, B and C of a triangle ABC intersect its circumcircle at points D, E and F respectively. Prove that angle EDF = 90 ^{o} â€“ Â½ âˆ A **

**Solution:**

Join ED, EF and DF. Also join BF, FA, AE and EC.

âˆ EBF = âˆ ECF = âˆ EDF â€¦.. (i) [Angle in the same segment]

In cyclic quadrilateral AFBE,

âˆ EBF + âˆ EAF = 180^{o} â€¦â€¦ (ii)

Similarly in cyclic quadrilateral CEAF,

âˆ EAF + âˆ ECF = 180^{o} â€¦â€¦. (iii)

Adding (ii) and (iii) we get,

âˆ EBF + âˆ ECF + 2âˆ EAF = 360^{o}

âˆ EDF + âˆ EDF + 2âˆ EAF = 360^{o} [From (i)]

âˆ EDF + âˆ EAF = 180^{o}

âˆ EDF + âˆ 1 + âˆ BAC + âˆ 2 = 180^{o}

But, âˆ 1 = âˆ 3 and âˆ 2 and âˆ 4 [Angles in the same segment]

âˆ EDF + âˆ 3 + âˆ BAC + âˆ 4 = 180^{o}

But, âˆ 4 = Â½ âˆ C, âˆ 3 = Â½ âˆ B

Thus, âˆ EDF + Â½ âˆ B + âˆ BAC + Â½ âˆ C = 180^{o}

âˆ EDF + Â½ âˆ B + 2 x Â½ âˆ A + Â½ âˆ C = 180^{o}

âˆ EDF + Â½ (âˆ A + âˆ B + âˆ C) + Â½ âˆ A = 180^{o}

âˆ EDF + Â½ (180^{o}) + Â½ âˆ A = 180^{o}

âˆ EDF + 90^{o} + Â½ âˆ A = 180^{o}

âˆ EDF = 180^{o} â€“ (90^{o} + Â½ âˆ A)

âˆ EDF = 90^{o} â€“ Â½ âˆ A

**11. In the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. IfÂ âˆ C = 20 ^{o}, find angle AOD.**

**Solution: **

Join OB.

In âˆ†OBC, we have

BC = OD = OB [Radii of the same circle]

âˆ BOC = âˆ BCO = 20^{o}

And ext. âˆ ABO = âˆ BCO + âˆ BOC

Ext. âˆ ABO = 20^{o} + 20^{o} = 40^{o} â€¦.. (1)

Now in âˆ†OAB,

OA = OB [Radii of the same circle]

âˆ OAB = âˆ OBA = 40^{o} [from (1)]

âˆ AOB = 180^{o} â€“ 40^{o} â€“ 40^{o} = 100^{o}

As DOC is a straight line,

âˆ AOD + âˆ AOB + âˆ BOC = 180^{o}

âˆ AOD + 100^{o} + 20^{o} = 180^{o}

âˆ AOD = 180^{o} â€“ 120^{o}

Thus, âˆ AOD = 60^{o}

**12. Prove that the perimeter of a right triangle is equal to the sum of the diameter of its incircle and twice the diameter of its circumcircle.**

**Solution:**

Letâ€™s join OL, OM and ON.

And, let D and d be the diameter of the circumcircle and incircle.

Also, let R and r be the radius of the circumcircle and incircle.

Now, in circumcircle ofÂ âˆ†ABC,

âˆ B = 90^{o}

Thus, AC is the diameter of the circumcircle i.e. AC = D

Let the radius of the incircle be â€˜râ€™

OL = OM = ON = r

Now, from B, BL and BM are the tangents to the incircle.

So, BL = BM = r

Similarly,

AM = AN and CL = CN = R

[Tangents from the point outside the circle]Now,

AB + BC + CA = AM + BM + BL + CL + CA

= AN + r + r + CN + CA

= AN + CN + 2r + CA

= AC + AC + 2r

= 2AC + 2r

= 2D + d

– Hence Proved

**13. P is the midpoint of an arc APB of a circle. Prove that the tangent drawn at P will be parallel to the chord AB.**

**Solution: **

First join AP and BP.

As TPS is a tangent and PA is the chord of the circle.

âˆ BPT = âˆ PABÂ [Angles in alternate segments]

But,

âˆ PBA = âˆ PAB [Since PA = PB]

Thus, âˆ BPT = âˆ PBA

But these are alternate angles,

Hence, TPS || AB

**14. In the given figure, MN is the common chord of two intersecting circles and AB is their common tangent.**

**Prove that the line NM produced bisects AB at P.**

**Solution: **

From P, AP is the tangent and PMN is the secant for first circle.

AP^{2} = PM x PN â€¦. (1)

Again from P, PB is the tangent and PMN is the secant for second circle.

PB^{2} = PM x PN â€¦. (2)

From (i) and (ii), we have

AP^{2} = PB^{2}

AP = PB

Thus, P is the midpoint of AB.

**15. In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. IfÂ âˆ DCQ = 40 ^{o} andÂ âˆ ABD = 60^{o}, find:**

**i)Â âˆ DBC**

**ii)Â âˆ BCP**

**iii)Â âˆ ADB**

**Solution:**

PQ is a tangent and CD is a chord.

âˆ DCQ = âˆ DBC [Angles in the alternate segment]

âˆ DBC = 40^{o} [As âˆ DCQ = 40^{o}]

(ii) âˆ DCQ + âˆ DCB + âˆ BCP = 180^{o}

40^{o} + 90^{o} + âˆ BCP = 180^{o} [As âˆ DCB = 90^{o}]

âˆ BCP = 180^{o} â€“ 130^{o} = 50^{o}

(iii) In âˆ†ABD,

âˆ BAD = 90^{o} [Angle in a semi-circle], âˆ ABD = 60^{o} [Given]

âˆ ADB = 180^{o} â€“ (90^{o} + 60^{o})

âˆ ADB = 180^{o} â€“ 150^{o} = 30^{o}

**16. The given figure shows a circle with centre O and BCD is a tangent to it at C. Show that:Â âˆ ACD + âˆ BAC = 90 ^{o}**

**Solution: **

Letâ€™s join OC.

BCD is the tangent and OC is the radius.

As, OC âŠ¥ BD

âˆ OCD = 90^{o}

âˆ OCD + âˆ ACD = 90^{o} â€¦. (i)

But, in âˆ†OCA

OA = OC [Radii of the same circle]

Thus, âˆ OCA = âˆ OAC

Substituting in (i), we get

âˆ OAC + âˆ ACD = 90^{o}

Hence, âˆ BAC + âˆ ACD = 90^{o}

**17. ABC is a right triangle with angle B = 90Âº. A circle with BC as diameter meets by hypotenuse AC at point D. Prove that: **

**i) AC x AD = AB ^{2}**

**ii) BD ^{2}Â = AD x DC.**

**Solution: **

i) InÂ âˆ†ABC, we have

âˆ B = 90^{o}Â and BC is the diameter of the circle.

Hence, AB is the tangent to the circle at B.

Now, as AB is tangent and ADC is the secant we have

AB^{2} = AD x AC

ii) InÂ âˆ†ADB,

âˆ D = 90^{o}

So, âˆ A + âˆ ABD = 90^{o} â€¦â€¦ (i)

But in âˆ†ABC, âˆ B = 90^{o}

âˆ A + âˆ C = 90^{o}Â â€¦â€¦. (ii)

From (i) and (ii),

âˆ C = âˆ ABD

Now in âˆ†ABD and âˆ†CBD, we have

âˆ BDA = âˆ BDC = 90^{o}

âˆ ABD = âˆ BCD

Hence, âˆ†ABD ~ âˆ†CBD by AA postulate

So, we have

BD/DC = AD/BD

Therefore,

BD^{2} = AD x DC

**18. In the given figure, AC = AE.**

**Show that:**

**i) CP = EP**

**ii) BP = DP**

**Solution:**

InÂ âˆ†ADC and âˆ†ABE,

âˆ ACD = âˆ AEBÂ [Angles in the same segment]

AC = AE [Given]

âˆ A = âˆ A [Common]

Hence, âˆ†ADC â‰… âˆ†ABE by ASA postulate

So, by C.P.C.T we have

AB = AD

But, AC = AE [Given]

So, AC â€“ AB = AE â€“ AD

BC = DE

InÂ âˆ†BPC and âˆ†DPE,

âˆ C = âˆ EÂ [Angles in the same segment]

BC = DE

âˆ CBP = âˆ CDE [Angles in the same segment]

Hence, âˆ†BPC â‰… âˆ†DPE by ASA postulate

So, by C.P.C.T we have

BP = DP and CP = PE

**19. ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = 120 ^{o}**

**Calculate:**

**i)Â âˆ BEC**

**ii) âˆ BED**

**Solution:**

i) Join OC and OB.

AB = BC = CD and âˆ ABC = 120^{o} [Given]

So, âˆ BCD = âˆ ABC = 120^{o}

OB and OC are the bisectors ofÂ âˆ ABC and âˆ BCDÂ andÂ respectively.

So, âˆ OBC = âˆ BCO = 60^{o}

InÂ âˆ†BOC,

âˆ BOC = 180^{o} â€“ (âˆ OBC + âˆ BOC)

âˆ BOC = 180^{o} â€“ (60^{o} + 60^{o}) = 180^{o} â€“ 120^{o}

âˆ BOC = 60^{o}

Arc BC subtendsÂ âˆ BOCÂ at the centre andÂ âˆ BECÂ at the remaining part of the circle.

âˆ BEC = Â½ âˆ BOC = Â½ x 60^{o}Â = 30^{o}

ii) In cyclic quadrilateral BCDE, we have

âˆ BED + âˆ BCD = 180^{o}

âˆ BED + 120^{o} = 180^{o}Â

Thus, âˆ BED = 60^{o}

**20. In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30 ^{o}, find:**

**(i)Â angle BCO**

**(ii)Â angle AOB**

**(iii)Â angle APB**

**Solution:**

In the given fig, O is the centre of the circle and, CA and CB are the tangents to the circle from C. Also,Â âˆ ACO = 30^{o}

P is any point on the circle. P and B are joined.

To find:

(i) âˆ BCOÂ

(ii)Â âˆ AOB

(iii) âˆ APB

Proof:

(i) In âˆ†OAC and âˆ†OBC, we have

OC = OC [Common]

OA = OB [Radii of the same circle]

CA = CB [Tangents to the circle]

Hence, âˆ†OAC â‰… âˆ†OBC by SSS congruence criterion

Thus, âˆ ACO = âˆ BCO = 30^{o}

(ii) As âˆ ACB = 30^{o} + 30^{o} = 60^{o}

And, âˆ AOB + âˆ ACB = 180^{o}

âˆ AOB + 60^{o} = 180^{o}

âˆ AOB = 180^{o} â€“ 60^{o}

âˆ AOB = 120^{o}

(iii) Arc AB subtends âˆ AOB at the center and âˆ APB is the remaining part of the circle.

âˆ APB = Â½ âˆ AOB = Â½ x 120^{o} = 60^{o}

**21. ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centers. Find the radii of the three circles.**

**Solution: **

Given: ABC is a triangle with AB = 10 cm, BC= 8 cm, AC = 6 cm. Three circles are drawn with centre A, B and C touch each other at P, Q and R respectively.

So, we need to find the radii of the three circles.

Let,

PA = AQ = x

QC = CR = y

RB = BP = z

So, we have

x + z = 10 â€¦.. (i)

z + y = 8 â€¦â€¦ (ii)

y + x = 6 â€¦â€¦. (iii)

Adding all the three equations, we have

2(x + y + z) = 24

x + y + z = 24/2 = 12 â€¦.. (iv)

Subtracting (i), (ii) and (iii) from (iv) we get

y = 12 â€“ 10 = 2

x = 12 â€“ 8 = 4

z = 12 â€“ 6 = 6

Thus, radii of the three circles are 2 cm, 4 cm and 6 cm.