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Circles Class 10 Maths NCERT Solutions Chapter 10 is concerned with the understanding of various aspects of circles. Circles are essential in mathematics because they are covered extensively in the geometry curriculum. Students preparing for their Class 10 exams will be able to clear all their concepts on circles at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all major concepts in detail, allowing students to understand the ideas better.

Circles Class 10 NCERT Solutions Circles, the tenth chapter of the section, concentrates on the essential concepts such as the tangents, the lengths of tangents, normal to a circle, the radius-tangent relationship, and so on. The properties based on the above mentioned topics are used to solve questions related to these topics step by step. All of these solutions are designed with the new CBSE pattern in mind so that students have a complete understanding of their tests.

Circle Class 10 Chapter 10 Questions and Answers are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing an in-depth analysis of the subjects. The Chapter 10 Maths Class 10 NCERT Solutions are available in pdf format below, and some of them are also included in the exercises.

Table of Content

Exercise 10.1

Question 1

How many tangents can be drawn at any point on the circle?

Solution

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Let P be any point on the circle with center $O$

$OP=radius$

Take a line L through P and Q as shown if L is perpendicular to OP

$OQ⊥OP$ because the perpendicular distance is shortest.

Every point except P lies outside the circle and line $l$ must be a tangent. At any given point one and only one tangent can be drawn.

Question 2

Fill in the Blanks:

(i) A tangent to a circle intersects it in ________ point(s).

(ii) A line intersecting a circle in two points is called a _______.

(iii) A circle can have _______ parallel tangents at the most.

(iv) A common point of a tangent to a circle and the circle is called ______.

Solution

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(i) One point

Circle is the locus of points equidistant from a given point, the center of the circle, and nd Tangent is the line which intersect circle at (ii) Secant

A line which intersect circle in two points is called a chord if this line passes through center then it is called **secant. (Points are A and B)**

(iii) Two

As circle has infinite points, so there will be infinite tangents can be drawn on these points which touches at only one point. (P and Q)

So there will be infinite pairs of tangents which are parallel.

(iv) Point of Contact

The common point of a tangent to a circle and the circle is called point of Question 3

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q, so that $OQ=12$ cm. Length of PQ is :

Solution

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Given,

Radius OP=$5$cm and OQ=$12$cm

PQ is the tangent to the circle.

$∠OPQ=90_{0}$

So,by Pythagoras theorem we get,

$PQ_{2}=OQ_{2}−OP_{2}$

$=>PQ_{2}=12_{2}−5_{2}$

$=>PQ_{2}=144−25$

$=>PQ_{2}=119$

$=>PQ=119 $cm

Question 4

Draw a circle and two lines parallel to a given line that one is a tangent and the other, a secant to the circle

Solution

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Let the circle has a center $O$ and $AB$ be a given line such that

Let $AB∥CD∥EF$.

Where $CD$ is a second ans $EF$ is a tangent intersecting the circle at $R$.

Exercise: 10.2

Question 1

From a point $Q$, the length of the tangent to a circle is $24$ cm and the distance of $Q$ from the centre is $25$ cm. The radius of the circle is

Solution

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Tangent=$xy$

point of contact=$b$

length of the tangent to a circle =$24cm$

i.e.,

$PQ=24cm$

$OQ=25cm$

prove

radius of circle i.e., OP

Proof

Since $xy$ is tangent

$OP⊥XY$

so, $∠OPQ=90$

So,

$△OPQ$ is a right angled triangle using by pythagoras

theorem

$OQ_{2}=OP_{2}+PQ_{2}$

$=>25_{2}=OP_{2}+24_{2}$

$=>OP_{2}=625−576$

$=>OP_{2}=49$

$=>OP=7$

Hence ,

radius=$OP=7cm$

Question 2

If $TP$ and $TQ$ are two tangents to a circle with centre $O$ so that $∠POQ=110_{o}$, then $∠PTQ$ is equal to

Solution

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Given, $∠POQ=110_{∘}$

We know,

$∠OPT=∠OQT=90_{∘}$ (Angle between the tangent and the radial line at the point of intersection of the tangent at the circle)

Now, in quadrilateral $POQT$

Sum of angles$=360$

$∠OPT+∠OQT+∠PTQ+∠POQ=360_{∘}$

$90+90+∠PTQ+110=360$

$∠PTQ=360−290$

$∠PTQ=70_{∘}$

We know,

$∠OPT=∠OQT=90_{∘}$ (Angle between the tangent and the radial line at the point of intersection of the tangent at the circle)

Now, in quadrilateral $POQT$

Sum of angles$=360$

$∠OPT+∠OQT+∠PTQ+∠POQ=360_{∘}$

$90+90+∠PTQ+110=360$

$∠PTQ=360−290$

$∠PTQ=70_{∘}$

Question 3

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of $80_{∘}$, then $∠POA$ is equal to:

Solution

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Given that,

PA and PB are two tangents a circle and $∠APB=80_{0}$

To find that $∠POA=?$

Construction:- join $OA,OBandOP$

Proof:- Since $OA⊥PA$ and $OB⊥PB$

Then $∠OAP=90_{0}$ and $∠OBP=90_{0}$

In

$ΔOAP&ΔOBP$

$OA=OB(radius)$

$OP=OP(Common)$

$PA=PB(lengthsoftangentdrawnfromexternalpointisequal)$

$∴ΔOAP≅ΔOBP(SSScongruency)$

So,

$[∠OPA=∠OPB(byCPCT)]$

So,

$∠OPA=21 ∠APB$

$=21 ×80_{0}=40_{0}$

In $ΔOPA,$

$∠POA+∠OPA+∠OAP=180_{0}$

$∠POA+40_{0}+90_{0}=180_{0}$

$∠POA+130_{0}=180_{0}$

$∠POA=180_{0}−130_{0}$

$∠POA=50_{0}$

The value of $∠POA$ is $50_{0}.$

Question 4

Prove that the tangents drawn at the end of a diameter of a circle are parallel

Solution

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To prove: PQ$∣∣$ RS

Given: A circle with centre O and diameter AB. Let PQ be the tangent at point A & Rs be the point B.

Proof: Since PQ is a tangent at point A.

OA$⊥$ PQ(Tangent at any point of circle is perpendicular to the radius through point of contact).

$∠OQP=90_{o}$ …………$(1)$

OB$⊥$ RS

$∠OBS=90_{o}$ ……………$(2)$

From $(1)$ & $(2)$

$∠OAP=∠OBS$

i.e., $∠BAP=∠ABS$

for lines PQ & RS and transversal AB

$∠BAP=∠ABS$ i.e., both alternate angles are equal.

So, lines are parallel.

$$\therefore PQ||RS.

Question 5

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Solution

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Given a circle with center $O$ and $AB$ the tangent intersecting circle at point $P$

and prove that $OP⊥AB$

We know that tangent of the circle is perpendicular to radius at points of contact Hence

$OP⊥AB$

So, $∠OPB=90_{o}..........(i)$

Now lets assume some point $X$

Such that $XP⊥AN$

Hence $∠XPB=90_{o}.........(ii)$

From eq $(i)$ & $(ii)$

$∠OPB=∠XPB=90_{o}$

Which is possible only if line $XP$ passes though $O$

Hence perpendicular to tangent passes though centre

Question 6

The length of a tangent from a point A at distance $5$ cm from the centre of circle is $4$ cm. The radius of the circle is

Solution

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Let $AT$ be the tangent drawn from a point $A$ to a circle with centre

$O$ and $OA=5$ cm and $AT=4$ cm. Since tangent at a point is

perpendicular to the radius through the point of contact

$∴OT⊥AT$

$∴$ from right angled $△OAT$,

$(OA)_{2}=(OT)_{2}+(TA)_{2}$

$⇒(5)_{2}=(OT)_{2}+(4)_{2}$

$⇒25−16=(OT)_{2}$

$⇒9=(OT)_{2}$

$⇒OT=3$ cm

$∴$ radius of the circle $=3$ cm.

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Practice more questions

A circle is a two-dimensional shape formed by all points in a plane that are equidistant from a specific point, called the center.

The tangent line to a plane curve at a given location in geometry is the straight line that "almost touches" the curve at that point. When the two endpoints of a chord coincide, the tangent to a circle is a specific case of the secant. A tangent to a circle is a line that only touches the circle once.

Few properties of tangents to a circle:

- A circle's tangent is perpendicular to the radius via the point of contact.
- The lengths of the two tangents to a circle from an exterior point are equal.

- A tangent to a circle from within the circle does not exist.
- A single tangent to a circle exists from a single point on the circle.
- There are two possible tangents to a circle from a point outside the circle.

Related Chapters

- Chapter 1 : Real NumbersChapter 2 : PolynomialsChapter 3 : Pair of Linear Equations in Two VariablesChapter 4 : Quadratic EquationsChapter 5 : Arithmetic ProgressionChapter 6 : TrianglesChapter 7 : Coordinate GeometryChapter 8 : Introduction to TrigonometryChapter 9 : Some Applications of TrigonometryChapter 11 : ConstructionsChapter 12 : Area Related to CirclesChapter 13 : Surface Areas and VolumesChapter 14 : StatisticsChapter 15 : Probability

Q1. What are the most important topics in NCERT Class 10 Chapter 10 Circles?

Answer: NCERT Solutions for Class 10 Maths Chapter 10 covers the following topics: introduction to circles, tangent to a circle, lengths of tangents, number of tangents from a point on a circle, the radius-tangent relationship, and the chapter summary.

Q2. How many exercises are there in Chapter 10 of Class 10 Maths?

Answer: NCERT Solutions Class 10 Maths Chapter 10 Circles contains a total of 17 questions separated into 2 exercises. Class 10 Maths Chapter 10 Circles contains a total of 17 sums, 10 of which are simple, 3 of which are fairly complicated, and 4 of which are tough.

Q3. How many tangents can a circle have?

Answer: Generally, there is no limit to the number of tangents a circle can have. It can be limitless since a circle is made up of infinite points that are all equally far from one another. Because there are infinite locations on the circumference of a circle, infinite tangents can be traced from them.

Q4. Do I have to practice all of the questions in NCERT Solutions Class 10 Maths Chapter 10 Circles?

Answer: All questions in NCERT Solutions Class 10 Maths Circles are based on key concepts that provide a comprehensive overview of the subject. Because the problems can be complex, practising all of the sums, as well as the examples, can help students learn the idea of circles and understand how to use various formulas. Students must also revise the given theorems and practice questions twice because the sums are proof-based.

Related Chapters

- Chapter 1 : Real NumbersChapter 2 : PolynomialsChapter 3 : Pair of Linear Equations in Two VariablesChapter 4 : Quadratic EquationsChapter 5 : Arithmetic ProgressionChapter 6 : TrianglesChapter 7 : Coordinate GeometryChapter 8 : Introduction to TrigonometryChapter 9 : Some Applications of TrigonometryChapter 11 : ConstructionsChapter 12 : Area Related to CirclesChapter 13 : Surface Areas and VolumesChapter 14 : StatisticsChapter 15 : Probability