Most Important Questions Class 10 Maths Ch 10 Circles with Solutions

Mathematics, often seen as a tough subject, can become manageable and even enjoyable with the right strategies and practice. Among the various chapters in Class 10 Math, Circles not only plays a vital role in the exam but also builds the foundation for more advanced topics in geometry. Solving the important Circles Class 10 questions is essential to excel in this chapter. These questions help students learn the fundamental concepts and develop a deeper understanding of the subject.

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Chapter 10 Circles: Important Questions

1. In the figure below, ΔPXY is formed using three tangents to a circle centred at O.

(Note: The figure is not to scale.)

Based on the construction, the sum of the tangents PA and PB is the perimeter of ΔPXY.

a. lesser than 

b. greater than 

c. equal to

d. (cannot be answered without knowing the tangent lengths)

Soln.

(c) equal to

Explanation:

As PA, PB and XY are the tangents. So, XA, XM, YM, and YB are also tangents because these are the line segments of the whole tangents.

And we know that if from one external point, two tangents are drawn to a circle, then they have equal tangent segments. Therefore,

XA=XM             .....(1)

YM=YB             .....(2)

Now, the perimeter of ∆PXY is=PX+XY+YP

=PX+(XM+YM)+YP

 =PX+(XA+YB)+YP     (From equation 1 and 2)

=(PX+XA)+(YB+YP)

=PA+PB          (From the figure)

Hence, the sum of the tangents PA and PB is equal to the perimeter of ∆PXY.

2. A circle has a centre O and radii OQ and OR. Two tangents, PQ and PR, are drawn from an external point, P.

In addition to the above information, which of these must also be known to conclude that the quadrilateral PQOR is a square?

i) OQ and OR are at an angle of 90°.

ii) The tangents meet at an angle of 90°.

a. only i) 

b. only ii) 

c. either i) or ii)

d. both i) and ii)

Soln. 

(c) either i) or ii)

Explanation:

We have to find the conditions which conclude that the quadrilateral PQOR is a square.

We know that a square has four equal sides and each angle as right angle.

We also know that the tangent to the circle is perpendicular to the radius of the circle at the point of contact.

So, ∠OQP=ORP=90°.

Now, if third angle is also 90°, then the fourth angle will be 90° because the sum of angles of a quadrilateral is 360°. Therefore,

If ∠QOR=90° (when radii are at angle of 90°) or ∠QPR=90° (when the tangents meet an angle of 90°), then the quadrilateral PQOR is a square.

Hence, either (i) or (ii) information must be known.

3. ABCD is a square. CD is a tangent to the circle with centre O as shown in the figure below.

(Note: The figure is not to scale.)

If OD = CE, what is the ratio of the area of the circle and the area of the square?

Show your steps and give valid reasons.

Soln. 

Given, A circle with centre O is drawn such that CD is the tangent to the circle such that OD=CE

We know that

Area of the square =s²
Area of circle =πr²

Let the side of square is 's’ and 'r' be the radius of circle.

In triangle DOC:

OD²+DC²=OC²

⇒r²+s²=2r²⇒r²+s²=4r²⇒s²=3r²⇒s=3r

∴Area of circle/Area of square=πr²/s²

Substitute the value of s in the formula.

⇒πr²3r²=π³

Hence, the required ratio is π:3.

4. Shown below is a circle with centre O and radius 5 units. PM and PN are tangents, and the length of chord MN is 6 units.

(Note: The figure is not to scale.)

Find the length of (PM + PN). Show your work.

Soln.

Connect P to O intersecting MN at T.

We know that OT bisects the chord MN at T.

∴ MT=NT=3 cm

In ∆OTM, 

OM²=MT²+OT²

OT²=5²-3²

OT²=25-9

OT²=16

OT=4 units

Let us consider PM=x units, PT=y units

In ∆PTM, 

PM²=PT²+TM²

x²=y²+3²

x²-y²=9

In ∆PMO, 

PO²=PM²+MO²

⇒x²+25=y+4²

⇒x²+25=y²+8y+16

⇒x²-y²=8y-9

Substitute the value of x²-y², we get

⇒9=8y-9

⇒8y=18

∴y=9/4 units

Substitute y to find the value for x

x²-94²=9

x²=9+8116

x²=22516

⇒x=154

We know that two tangents drawn through an external point are equal.

⇒PM=PN

∴PM+PN

=154+154=304

⇒PM+PN

=7.5 units

Hence, the required answer is 7.5 units.

5. Shown below is a circle with centre O. VP = 34 cm, PR and S are the points of tangency.

(Note: The figure is not to scale.)

Find the area of the shaded region in terms of π. Show your steps and give valid reasons.

Soln. 

Given, VP = 34 cm, PR = 36 cm and RS = 17 cm.

We know that if two tangents are drawn to a circle from an external point then they are equal.

∴RS=RQ

⇒RS=RQ=17 cm

⇒PQ=PR-QR

=36 -17

∴PQ=19 cm

PQ and PW are the tangents drawn from an external point. So, they are equal.

⇒PQ=PW

=19 cm

VW and VU are the tangents drawn from an external point. So, they are equal.

⇒VW=VP-PW

=34cm-19cm

⇒VW=15cm

∴VW=VU=15cm

We know that radius is perpendicular to the tangent. So,∠VWO=∠VUO=90°

∴VWOU is a square.

⇒VW=WO=OU=UV=15 cm

Area of a semicircle =πr²/2

⇒π×15×15/2

=112.5π cm²

Hence, the required answer is 112.5π cm².

6. In the figure below, a circle with centre O is inscribed inside ΔLMN. A and B are the points of tangency.

(Note: The figure is not to scale.)

(Note: The figure is not to scale.)

Find ∠ANB. Show your steps.

Soln. 

Given, major∠AOB=260°

Minor ∠AOB=360°-260°=100°

We know that radius and tangent are perpendicular. So, 

∠NAO=∠NBO=90°

In a quadrilateral NAOB,

 ∠NAO+∠AOB+∠NBO+∠ANB=360°

90°+100°+90°+∠ANB=360°

∠ANB=360°-280°

⇒∠ANB=80°

Hence, the value of ∠ANB is 80°.

7. A point is 25 cm from the centre of a circle of radius 15 cm.

Find the length of the tangent from the point to the circle. Show your steps.

Soln. 

Given, radius =15cm

Let OB be radius and PB is a tangent.

Now we know that radius is perpendicular to point of contact

OB is perpendicular to PB

Hence ∠PBO=90°

Consider ∆PBO

Using Pythagoras theorem

PB²+OB²=PO²

PB²+15²=25²

PB²+225=625

PB²=625-225

PB²=400

PB=√400

PB=20 cm

Hence, the length of the tangent from the point to the circle is 20 cm.

8. In the figure below, PQ and RQ are tangents to the circle with centre O and radius 6√3 cm.

(Note: The figure is not to scale.)

i) Prove that APQR is an equilateral triangle.

ii) Find the length of RP. Show your steps along with a diagram and give valid reasons.

Soln. 

Given that PQ and RQ are tangents drawn from an external point Q and ∠PQR=60°

We know that the lengths of the tangents drawn from an external point to a circle are equal.

∴PQ=RQ

⇒△PQR is an isosceles triangle.

Let ∠QPR=∠QRP=x°

∴∠PQR+∠QPR+∠QRP=180°

⇒60°+x+x=180°⇒2x=120°

⇒x=60°

All the angles are equal to 60°.

∴△PQR is an equilateral triangle.

ii) To find the length of RP we join OP and OQ

We know that radius is perpendicular to tangent. So, ∠OPQ=90°

⇒∠OPS=∠OPQ-∠QPR

⇒∠OPS=90-60=30°

Given that radius is 6 cm

In △OPS, 

cos 30°=PS/OP

√3/2=PS/6

⇒PS=33 cm

We know that the perpendicular from the centre of a circle to a chord bisects the chord. So, RP=2 PS=63 cm.

9. Shown below is an ΔPQR inscribed in a semicircle.

A circle is drawn such that QR is tangent to it at the point R.

How many such circles can be drawn? Justify your answer

Soln. 

Infinite circles can be drawn such that QR is a tangent at point R.

We know that ∠PRQ = 90° because it is the angle in a semicircle and also the radius is perpendicular to the tangent at the point of contact.

∴ Infinite circles can be drawn with their radii lying on extended PR and R being a point on the circumference of the circle .

10. In the figure below, O is the centre of two concentric circles. APQR is an equilateral triangle such that its vertices and sides touch the bigger and smaller circles respectively. The difference between the area of the bigger circle and the smaller circle is 616 cm².

(Note: The figure is not to scale.)

Find the perimeter of APQR. Draw a rough diagram, show your work and give reasons.

(Note: Take π as 22/7)

Soln. 

Join from centre O to points P and S.

∴ πOP²-OS²=616 cm²

⇒OP²-OS²

=616π

=196 cm²

In ∆OPS, using Pythagoras' theorem, we get

OP²-OS²=PS²

∴PS²=196

PS=14 cm

We know that a perpendicular from the centre of a circle bisects th chord.

⇒PR=2×PS=28 cm

Given that ∆PQR is an equilateral triangle. 

Perimeter =3×side length

⇒3×28

⇒84 cm

Hence, the required perimeter is 84 cm.

What are circles, and why should you use circles in class 10 important questions?

The chapter on circles in Class 10 introduces students to several essential geometric concepts. These include the different parts of a circle, such as the radius, diameter, chord, and tangent, as well as the relationships between these components. The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact, which is one of the chapter's most important theorems.

Solving Class 10 circle important questions becomes important. Regular practice helps students understand how to apply theoretical knowledge in practical situations, building confidence for exams.

• Circle is a chapter that often carries a favorable weight in Class 10 board exams. By focusing on important questions in Class 10, students can target high-scoring areas. These questions frequently appear in exams, so practicing them regularly ensures that students are well-prepared. Solving extra questions for Class 10 math circles with solutions helps students become familiar with the types of problems that may appear in the exam.
• A few key theorems, such as the one about tangents, center the chapter. These theorems form the backbone of all questions related to circles. By practicing the extra circles' questions from Class 10, students can apply these theorems in various ways, ensuring a thorough understanding.
• Geometry, particularly circles, requires a high level of problem-solving ability. The more you practice, the better you get at recognizing patterns and solving questions efficiently. By working through circles on important questions in Class 10, students can develop their problem-solving skills. 
• Some students may find geometry confusing, especially when it comes to complex problems involving tangents and chords. However, by solving additional Class 10 circle questions, students gain the confidence to tackle even the most challenging problems. As you solve more questions, you begin to see connections between different concepts, making it easier to understand and solve complex problems during the exam.
• Many students struggle with time management during exams, especially when faced with lengthy geometry problems. By practicing circles for important questions in Class 10, students learn to solve problems more efficiently, which can help them manage their time better during exams.  Regular practice also helps students identify shortcuts and quick ways to solve without wasting time.

How to Effectively Use Class 10 Circles Important Questions?

Solving Circles Class 10's important questions is essential, but using them effectively is equally important. Here are some strategies to maximize your learning and preparation:

• Before diving into complex problems, make sure you thoroughly understand the basics of circles. Review important concepts such as the parts of a circle, properties of tangents, and key theorems. Without a solid foundation, it’s difficult to tackle more challenging problems. Once you're confident with the fundamentals, you can move on to Class 10 circles with extra questions that require deeper understanding.
• It’s tempting to start with the hardest questions, but that can lead to frustration if you're not fully prepared. Instead, start with simpler problems to build your confidence. Gradually move on to circle important questions in Class 10 that are more challenging. This approach helps you progress naturally, ensuring that you don’t get overwhelmed.
• Don’t limit yourself to just one type of problem. The chapter on circles offers a wide range of problems, from basic identification of parts of a circle to complex applications of tangents and chords. By practicing a variety of Class 10 circle important questions, you expose yourself to different types of challenges, which makes you more adaptable and better prepared for the exam.
• Geometry is a visual subject, and solving problems without drawing diagrams can make it more difficult. Always draw neat and accurate diagrams for each problem. Whether it's marking tangents or drawing radii, visualizing the problem helps you understand the relationships between different elements of the circle. Diagrams can often provide clues to solving the problem more effectively.
• One of the best ways to prepare for exams is by taking mock tests. Set a timer and work through a set of additional questions for Class 10 math circles, each with solutions provided under exam conditions. This will not only help you practice but also improve your time management skills. By simulating the exam environment, you’ll be better prepared to handle pressure and complete the paper on time.

• Geometry involves a lot of theorems and properties. Create a concise formula sheet that lists all the important theorems related to circles. Keep this sheet handy while practicing, and refer to it regularly to reinforce your memory.

Circles Class 10 Important Questions are an invaluable tool in preparing for your board exams. With a solid understanding of the fundamental concepts and consistent practice of extra questions for Class 10 math circles with solutions, students can build confidence, sharpen their problem-solving skills, and ensure success in their exams. Circles may seem like a challenging chapter at first, but with the right approach and a good mix of theory and practice, it can become one of your strongest subjects.

Remember, the key to mastering this chapter lies in regular practice and thorough revision.