## RS Aggarwal Class 10 Solutions Chapter 12 Circles Test Yourself

These Solutions are part of RS Aggarwal Solutions Class 10. Here we have given RS Aggarwal Solutions Class 10 Chapter 12 Circles Test Yourself.

**Other Exercises**

- RS Aggarwal Solutions Class 10 Chapter 12 Circles Ex 12A
- RS Aggarwal Solutions Class 10 Chapter 12 Circles Ex 12B
- RS Aggarwal Solutions Class 10 Chapter 12 Circles MCQS
- RS Aggarwal Solutions Class 10 Chapter 12 Circles Test Yourself

**MCQ**

**Question 1.**

**In the given figure, O is the centre of a circle, PQ is a chord and the tangent PT at P makes an angle of 50° with PQ. Then, ∠POQ = ?**

**(a) 130°**

**(b) 100°**

**(c) 90°**

**(d) 75°**

**Solution:**

In the given figure,

PT is the tangent and PQ is the chord of the circle with centre O.

∠OPT = 50°

OP is radius and PT is the tangent.

OP ⊥ PT ⇒ ∠OPT = 90°

∠OPQ + ∠QPT = 90°

⇒ ∠OPQ + 50° = 90°

⇒ ∠OPQ = 90° – 50° = 40°

In ∆OPQ,

OP = OQ (radii of the same circle)

∠OQP = ∠OPQ = 40°

In ∆OPQ,

∠POQ = 180° – (∠OPQ + ∠OQP)

= 180° – (40° + 40°) = 180° – 80°

= 100° (b)

**Question 2.**

**If the angle between two radii of a circle is 130°, then the angle between the tangents at the ends of the radii is**

**(a) 65°**

**(b) 40°**

**(c) 50°**

**(d) 90°**

**Solution:**

Angle between two radii of a circle = 130°

Then ∠APB = 180° – ∠AOB

= 180°- 130° = 50° (c)

**Question 3.**

**If tangents PA and PB from a point P to a circle with centre O are drawn so that ∠APB = 80°, then ∠POA = ?**

**(a) 40°**

**(b) 50°**

**(c) 80°**

**(d) 60°**

**Solution:**

In the given figure,

PA and PB are the tangents drawn from P to the circle with centre O

∠APB = 80°

OA is radius of the circle and AP is the tangent

OA ⊥ AP ⇒ ∠OAP = 90°

OP bisects ∠APB,

∠APO = x 80 = 40°

∠POA = 180° – (90° + 40°)

= 180° – 130° = 50° (b)

**Question 4.**

**In the given figure, AD and AE are the tangents to a circle with centre O and BC touches the circle at F. If AE = 5 cm, then perimeter of ∆ABC is**

**(a) 15 cm**

**(b) 10 cm**

**(c) 22.5 cm**

**(d) 20 cm**

**Solution:**

In the given figure, AD and AE are tangents to the circle with centre O.

BC is the tangent at F which meets AD at C and AE at B

AE = 5 cm

AE and AD are the tangents to the circle

AE = AD = 5 cm

Tangents from an external point drawn to the circle are equal

CD = CF and BE = BF

Now, perimeter of ∆ABC = AB + AC + BC

= AB + AC + BF + CF (BE = BF and CF = CD)

= AB + AC + BE + CD

= AB + BE + AC + CD

= AE + AD

= 5 + 5 = 10 cm (b)

**Short-Answer Questions**

**Question 5.**

**In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm, find x.**

**Solution:**

In the given figure, a quadrilateral ABCD is circumscribed a circle touching its sides at P, Q, R and S respectively.

AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm

A circle touches the sides of a quadrilateral ABCD.

AB + CD = BC + AD …(i)

Now, AP and AS are tangents to the circle

AP = AS = 5 cm …(ii)

Similarly, CQ = CR = 3 cm

BP = BQ = x – 5 = 4

BQ = BC – CQ = 7 – 3 = 4 cm

x – 5 = 4

⇒ x = 4 + 5 = 9cm

**Question 6.**

**In the given figure, PA and PB are the tangents to a circle with centre O. Show that the points A, O, B, P are concylic.**

**Solution:**

In the given figure, PA and PB are the tangents drawn from P to the circle.

OA and OB are the radii of the circle and AP and BP are the tangents.

OA ⊥ AP and OB ⊥ BP

∠OAP = ∠OBP = 90°

In quad. AOBP

∠A + ∠B = 90° + 90° = 180°

But these are opposite angles of a quadrilateral

AOBP is a cyclic quadrilateral

A, O, B, P are concyclic

**Question 7.**

**In the given figure, PA and PB are two tangents from an external point P to a circle with centre O. If ∠PBA = 65°, find ∠OAB and ∠APB.**

**Solution:**

In the given figure, PA and PB are two tangents to the circle with centre O from an external point P.

∠PBA = 65°,

To find : ∠OAB and ∠APB

In ∆APB

AP = BP (Tangents from P to the circle)

∠PAB = ∠PBA = 65°

∠APB = 180° – (∠PAB + ∠PBA)

= 180° – (65° + 65°) = 180° – 130° = 50°

OA is radius and AP is tangent

OA ⊥ AP

∠OAP = 90°

∠OAB = ∠OAP – ∠PAB = 90° – 65° = 25°

Hence, ∠OAB = 25° and ∠APB = 50°

**Question 8.**

**Two tangent segments BC and BD are drawn to a circle with centre O such that ∠CBD = 120°. Prove that OB = 2BC.**

**Solution:**

Given : In the figure,

BC and BD are the tangents drawn from B

to the circle with centre O.

∠CBD = 120°

To prove : OB = 2BC

Construction : Join OB.

Proof: OB bisects ∠CBD (OC = OD and BC = BD)

**Question 9.**

**Fill in the blanks.**

**(i) A line intersecting a circle in two distinct points is called a ……..**

**(ii) A circle can have …………. parallel tangents at the most.**

**(iii) The common point of a tangent to a circle and the circle is called the ………..**

**(iv) A circle can have …….. tangents.**

**Solution:**

(i) A line intersecting a circle in two distinct points is called a **secant.**

(ii) A circle can have **two** parallel tangents at the most.

(iii) The common point of a tangent to a circle and the circle is called the **point of contact.**

(iv) A circle can have infinitely **many** tangents.

**Question 10.**

**Prove that the lengths of two tangents drawn from an external point to a circle are equal.**

**Solution:**

Given : In a circle, from an external point P, PA and PB are the tangents drawn to the circle with centre O.

To prove : PA = PB

Construction : Join OA, OB and OP.

Proof : OA and OB are the radii of the circle and AP and BP are tangents.

OA ⊥ AP and OB ⊥ BP

⇒ ∠OAP = ∠OBP = 90°

Now, in right ∆OAP and ∆OBP,

Hyp. OP = OP (common)

Side OA = OB (radii of the same circle)

∆OAP = ∆OBP (RHS axiom)

PA = PB (c.p.c.t.)

Hence proved.

**Short-Answer Questions**

**Question 11.**

**Prove that the tangents drawn at the ends of the diameter of a circle are parallel.**

**Solution:**

Given : In a circle with centre O and AB is its diameter.

From A and B, PQ and RS are the tangents drawn to the circle

To prove : PQ || RS

Proof : OA is radius and PAQ is the tangent

OA ⊥ PQ

∠PAO = 90° …(i)

Similarly, OB is the radius and RBS is tangent

∠OBS = 90° …(ii)

From (i) and (ii)

∠PAO = ∠OBS

But there are alternate angles

PQ || RS

**Question 12.**

**In the given figure, if AB = AC, prove that BE = CE.**

**Solution:**

Given : In the given figure,

In ∆ABC,

AB = AC.

A circle is inscribed the triangle which touches it at D, E and F

To prove : BE = CE

Proof: AD and AF are the tangents drawn from A to the circle

AD = AF

But, AB = AC

AB – AD = AC -AF

⇒ BD = CF

But BD = BE and CF = CE (tangent drawn to the circle)

But BD = CF

BE = CE

Hence proved.

**Question 13.**

**If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the centre.**

**Solution:**

Given : In a circle from an external point P, PA and PB are the tangents to the circle

OP, OA and OB are joined.

To prove: ∠POA = ∠POB

Proof: OA and OB are the radii of the circle and PA and PB are the tangents to the circle

OA ⊥ AP and OB ⊥ BP

∠OAP = ∠OBP = 90°

Now, in right ∆OAP and ∆OBP,

Hyp. OP = OP (common)

Side OA = OB (radii of the same circle)

∆OAP = ∆OBP (RHS axiom)

∠POA = ∠POB (c.p.c.t.)

Hence proved.

**Question 14.**

**Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.**

**Solution:**

Given : A circle with centre O, PA and PB are the tangents drawn from A and B which meets at P.

AB is chord of the circle

To prove : ∠PAB = ∠PBA

Construction : Join OA, OB and OP

Proof: OA is radius and AP is tangent

OA ⊥ AP ⇒ ∠OAP = 90°

Similarly, OB ⊥ BP ⇒ ∠OBP = 90°

In ∆OAB, OA = OB (radii of the circle)

∠OAB = ∠OBA

⇒ ∠OAP – ∠OAB = ∠OBP – ∠OBA

⇒ ∠PAB = ∠PBA

Hence proved.

**Question 15.**

**Prove that the parallelogram circumscribing a circle, is a rhombus.**

**Solution:**

Given : A parallelogram ABCD is circumscribed a circle.

To prove : ABCD is a rhombus.

Proof: In a parallelogram ABCD.

Opposite sides are equal and parallel.

AB = CD and AD = BC

Tangents drawn from an external point of a circle to the circle are equal.

AP = AS BP = BQ

CQ = CR and DR = DS

Adding, we get

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

⇒ (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

⇒ AB + CD = AD + BC [AB = CD and AD = BC]

⇒ AB + AB = BC + BC

⇒ 2AB = 2BC

⇒ AB = BC

But AB = CD and BC = AD

AB = BC = CD = AD

Hence || gm ABCD is a rhombus.

**Question 16.**

**Two concentric circles are of radii 5 cm and 3 cm respectively. Find the length of the chord of the larger circle which touches the smaller circle.**

**Solution:**

Given : O is the centre of two concentric circles with radii 5 cm and 3 cm respectively.

AB is the chord of the larger circle which touches the smaller circle at P.

OP and OA are joined.

To find : Length of AB

Proof: OP is the radius of the smaller circle and touches the smaller circle at P

OP ⊥ AB and also bisects AB at P

AP = PB = AB

Now, in right ∆OAP,

OA² = OP² + AP² (Pythagoras Theorem)

⇒ (5)² = (3)² + AP²

⇒ 25 = 9 + AP²

⇒ AP² = 25 – 9 = 16 = (4)²

AP = 4 cm

Hence AB = 2 x AP = 2 x 4 = 8 cm

**Long-Answer Questions**

**Question 17.**

**A quadrilateral is drawn to circumscribed a circle. Prove that the sum of opposite sides are equal.**

**Solution:**

In the figure, quad. ABCD is circumscribed about a circle which touches its sides at P, Q, R and S respectively

To prove : AB + CD = AD + BC

Proof: Tangents drawn from an external point to a circle are equal

AP = AS

BP = BQ

CR = CQ

DR = DS

Adding, we get,

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

⇒ (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

⇒ AB + CD = AD + BC

Hence AB + CD = AD + BC

**Question 18.**

**Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.**

**Solution:**

Given : A quad. ABCD circumscribe a circle with centre O and touches at P, Q, R and S respectively

OA, OB, OC and OD are joined forming angles AOB, BOC, COD and DOA

To prove : ∠AOB + ∠COD = 180°

and ∠BOC + ∠AOD = 180°

Construction : Join OP, OQ, OR and OS

Proof: In right ∆AOP and ∆AOS,

Side OP = OS (radii of the same circle)

Hyp. OA = OA (common)

∆AOP = ∆AOS (RHS axiom)

∠1 = ∠2 (c.p.c.t.)

Similarly, we can prove that

∠4 = ∠3

∠5 = ∠6

∠8 = ∠7

Adding, we get

∠1 + ∠4 + ∠5 + ∠8 = ∠2 + ∠3 + ∠6 + ∠7

⇒ (∠1 + ∠8) + (∠4 + ∠5) = (∠2 + ∠3) + (∠6 + ∠7)

⇒ ∠AOB + ∠COD = ∠AOD + ∠BOC

But ∠AOB + ∠BOC + ∠COD + ∠DOA = 360° (angles at a point)

∠AOB + ∠COD = ∠AOD + ∠BOC = 180°

Hence proved

**Question 19.**

**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 segments joining the points of contact at the centre.**

**Solution:**

Given : From an external point P, PA and PB are the tangents drawn to the circle,

OA and OB are joined.

To prove : ∠APB + ∠AOB = 180°

Construction : Join OP.

Proof : Now, in ∆POA and ∆PBO,

OP = OP (common)

PA = PB (Tangents from P to the circle)

OA = OB (Radii of the same circle)

∆POA = ∆PBO (SSS axiom)

∠APO = ∠BPO (c.p.c.t.)

and ∠AOP = ∠BOP (c.p.c.t.)

OA and OB are the radii and PA and PB are the tangents

OA ⊥ AP and OB ⊥ BP

⇒ ∠OAP = 90° and ∠OBP = 90°

In ∆POA,

∠OAP = 90°

∠APO + ∠AOP = 90°

Similarly, ∠BPO + ∠BOP = 90°

Adding, we get

(∠APO + ∠BPO) + (∠AOP + ∠BOP) = 90° + 90°

⇒ ∠APB + ∠AOB = 180°.

Hence proved.

**Question 20.**

**PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.**

**Solution:**

Given : PQ is chord of a circle with centre O.

TP and TQ are tangents to the circle

Radius of the circle = 10 cm

i.e. OP = OQ = 10 cm and PQ = 16 cm

To find : The length of TP.

OT bisects the chord PQ at M at right angle.

PM = MQ = = 8 cm

In right ∆PMO,

OP² = PM² + MO² (Pythagoras Theorem)

⇒ (10)² = (8)² + MO²

⇒ 100 = 64 + MO²

⇒ MO² = 100 – 64 = 36 = (6)²

MO = 6 cm

Let TP = x and TM = y

In right ∆TPM,

TP² = TM² + PM²

⇒ x² = y² + 8²

⇒ x² = y² + 64 …(i)

and in right ∆TPM

OT² = TP² + OP²

⇒ (y + 6)² = x² + 10²

⇒ y² + 12y + 36 = x² + 100

⇒ y² + 12y + 36 = y2 + 64 + 100 {From (i)}

⇒ 12y = 64 + 100 – 36 = 128

Hope given RS Aggarwal Solutions Class 10 Chapter 12 Circles Test Yourself are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

## Leave a Reply