Circles: Class 10 Maths · Chapter 10

Mathematics · Companion notes, simulations and practice · Official NCERT chapter PDF ↗

Published 12 Oct 2026 · Updated 12 Oct 2026

50 min
1 simulation
25 practice questions
10 MCQs

What you will learn in Circles

Decide whether a line is a non-intersecting line, a secant or a tangent to a circle, by counting common points or by comparing its distance from the centre with the radius.
Describe a tangent as the limiting position of a secant, and explain why a circle has exactly one tangent at each point and at most two tangents parallel to a given line.
Prove that the radius to the point of contact is perpendicular to the tangent, and use this right angle with Pythagoras theorem to find lengths.
State how many tangents pass through a point inside a circle, on it and outside it.
Prove that the two tangents from an external point are equal, and that the line from that point to the centre bisects the angle between them.
Solve board-style problems on tangent lengths, angles between tangents, concentric circles, and triangles and quadrilaterals drawn around a circle.

Original companion text aligned to NCERT Mathematics Class 10, Chapter 10, pp. 144–153. Not affiliated with or endorsed by NCERT.

10.1 Introduction

Watch a kite flyer on Makar Sankranti letting out string. The thread unwinds from the round firki and leaves it in a straight line that only grazes the rim. Lines like that, which brush a circle at one point, are what this chapter is about.

So here is the first question: how many points can a straight line and a circle have in common? To answer it you need only the Class 9 picture of a circle (वृत्त): each of its points is exactly $r$ from the centre $O,$ where $r$ is the radius. Chords and arcs, which you met there, will turn up again along the way.

Lay a bangle flat on your desk and slide a ruler towards it, keeping the edge straight. Think of the edge as a line and the bangle as a circle (Fig. 10.1). Three things can happen, one after another:

The edge misses the bangle. Line and circle have no point in common. The line is a non-intersecting line for this circle.
The edge just grazes the bangle. They share exactly one point. The line is a tangent (स्पर्श रेखा) to the circle.
The edge cuts across the bangle. They share two points. The line is a secant (छेदक रेखा) of the circle.

2/3When $d = r$ the line meets the circle at one point $T :$ a tangent.

Fig. 10.1As a line moves towards a circle, the number of common points goes 0, 1, 2.NCERT Fig. 10.1 · p. 144

A circle with centre O and a radius r is drawn on the left, and a vertical line stands to its right. In the first step the line is far away: its distance d from O, shown by a dashed segment, is more than r, and the panel reads Misses, d > r, 0 points. In the second step the line has slid closer until d equals r; it touches the circle at one point T, and the panel reads Tangent, d = r, 1 point. In the third step the line has moved further in, d is less than r, it crosses the circle at A and B, and the panel reads Secant, d < r, 2 points.

Could a line meet a circle in three points, or more? No. In Class 9 you saw that three points on one straight line never lie on a circle. So a line and a circle share at most two points, and the three cases above are the only ones possible.

The figure also gives a quick test. Let $d$ be the perpendicular distance from the centre $O$ to the line, and $r$ the radius. If $d > r$ the line misses the circle; if $d = r$ it touches it; if $d < r$ it cuts it twice. You will see the reason in Section 10.2.

Quick check

1 of 3

A circle has radius 5 cm, and a line is 3 cm from its centre. What is the line?

a secanta tangenta non-intersecting lineit cannot be decided

Show answer

Answer (A) a secant

d = 3 < 5 = r,

so the line cuts the circle twice: a secant. A tangent needs

d = r

exactly.

10.2 Tangent to a Circle

Definition

Tangent

A tangent to a circle is a straight line that has exactly one point in common with the circle.

ExampleThe kite string at the point where it leaves the firki.

Two questions follow at once. Does every point of a circle have a tangent through it? And could one point have two different tangents? A small experiment answers both.

Activity 10.1NCERT Activity 1 · p. 145

Turn a straw about a point of a bangle

You’ll need

a bangle
a drinking straw
a pin and a small lump of clay
a sheet of paper and a pencil

Predict first

You pin the straw to the paper at a point

P

of the traced circle and turn it slowly through half a turn. In how many positions will the straw meet the circle only at

P

in no positionin exactly one positionin exactly two positionsin every position

Trace the inside of the bangle on the paper and mark a point $P$ on the circle.
Pin the middle of the straw to the paper at $P,$ with a lump of clay under the sheet to hold the pin. The straw can now turn about $P .$
Start with the straw lying across the circle. Mark the second point where it crosses the circle and call it $Q .$
Turn the straw a little at a time. Each time, mark the new position of $Q$ and measure the chord $P Q .$
Find the one position in which $Q$ disappears into $P .$ Draw the straw's line there, then keep turning and watch $Q$ appear on the other side.

Table 10.1

Turn	Is there a $Q$ ?	$P Q$ (cm)
1
2
3
4
5

Your entries are saved automatically.

4/4 $Q$ has merged with $P :$ the line meets the circle only at $P .$ It is the tangent at $P .$

Fig. 10.2As the line turns about

P,

the second point

Q

slides into

P .

NCERT Fig. 10.3 (i) · p. 145

A circle with centre O and a point P at its lowest point. A straight line passes through P. In the first step it rises steeply and cuts the circle again at a point Q far from P; the chord PQ is highlighted and the corner reads 2 points in common. In the second and third steps the line has turned towards the horizontal and Q has slid much closer to P, so the highlighted chord is shorter. In the fourth step the line is horizontal, Q has merged with P, the line is labelled tangent, and the corner reads Only P in common.

A second experiment looks at a whole family of parallel lines instead of lines through one point.

Activity 10.2NCERT Activity 2 · p. 146

Parallel lines on squared paper

You’ll need

a maths notebook with 1 cm squares, or graph paper
a compass
a ruler
a pencil

Predict first

You draw a circle and look at the chords cut by the horizontal grid lines, starting with the line through the centre and moving upwards. What happens to the chords?

they get longerthey stay the same lengththey get shorter but never vanishthey get shorter, then vanish

Draw a circle of radius 5 cm with its centre $O$ on a corner of a square. The horizontal grid lines are all parallel.
The grid line through $O$ is a secant. Measure the chord it cuts.
Move up one grid line at a time. Measure each chord and fill the table. The last column was worked out with Pythagoras, so you can check your drawing.
Repeat below $O .$ Which lines meet the circle at one point only? Which miss it?

Table 10.2Distance

d

of each line from

O,

and its chord, in cm

$d$	Measured	Calculated
0		10
1		9.8
2		9.2
3		8
4		6
5		0
6		none

Your entries are saved automatically.

The chords shrink as the lines move away from $O :$ about 10, 9.8, 9.2, 8 and 6 cm. The line 5 cm above $O$ touches the circle at a single point, and so does the line 5 cm below. Lines 6 cm away miss the circle. So out of the whole family of parallel lines, exactly two are tangents, one on each side of the centre (Fig. 10.3). Once again the chord of a secant has shrunk to nothing to give a tangent.

3/3At 5 cm from $O,$ on each side, the chord shrinks to a point: two tangents parallel to the grid lines.

Fig. 10.3On squared paper the chords shrink until two parallel lines just touch the circle.NCERT Fig. 10.3 (ii) · p. 146

Squared paper with 1 cm squares and a circle of radius 5 cm centred at O. In the first step the grid line through O is highlighted as a chord of length 10 cm. In the second step the chords on the grid lines 1, 2, 3 and 4 cm above and below O are highlighted too; labels on the right give 10, 9.8, 9.2, 8 and 6 cm for the upper half, so the chords shrink as the lines move away from O. In the third step the two lines 5 cm above and 5 cm below O are drawn as tangents touching the circle at one point each, with the secants between them.

Why these numbers? From Class 9, the perpendicular from the centre to a chord bisects the chord. So half the chord, the distance $d$ of the line from $O$ and the radius $r$ form a right triangle, and Pythagoras gives the length of the chord:

chord = 2 \sqrt{r^{2} - d^{2}}

(10.1)

where

$r$: the radius of the circle
$d$: the distance of the line from the centre

With $r = 5$ and $d = 3$ the chord is $2 \sqrt{16} = 8$ cm, as in the table. As $d$ grows towards $r$ the chord shrinks, and at $d = r$ it is zero: the line is a tangent. For $d > r$ the number under the root is negative, so there is no chord and the line misses the circle. This is the test from Section 10.1, now with a reason.

Definition

Point of contact

The single point that a tangent shares with the circle. We say the tangent touches the circle at this point.

Now join the centre to the point of contact. The radius you draw and the tangent seem to meet at a right angle. Check it with a protractor before we prove it.

Try thisNCERT Fig. 10.4 · p. 146

Measure the angle first

You’ll need

a compass
a ruler
a protractor

Predict first

You draw a tangent at a point

T

of a circle with centre

O

and join

O T .

What angle do

O T

and the tangent make?

about

45^{\circ}

about

60^{\circ}

it depends on where

T

isabout

90^{\circ}

Draw a circle of radius 4 cm with centre $O .$
Lay your ruler so that its edge just touches the circle, and draw the line. Mark the touching point $T .$
Join $O T$ and measure the angle between $O T$ and your line.
Repeat at two more points of the circle.

Table 10.3

Tangent	Angle (degrees)
1
2
3

Your entries are saved automatically.

Theorem 10.1Tangent and radius

The radius drawn to the point of contact of a tangent is perpendicular to that tangent.

Given: a circle with centre $O$ and radius $r,$ and a tangent $ℓ$ that touches it at $P .$

To prove: $O P ⊥ ℓ .$

Proof: First we show that $P$ is the point of $ℓ$ nearest to $O .$

Let $Q$ be a point of $ℓ$ different from $P .$ It cannot lie on the circle, because $P$ is the only point that $ℓ$ shares with the circle. It cannot lie inside the circle either: any line through an inside point leaves the circle twice, and $ℓ$ would be a secant. Hence $Q$ is outside the circle, and $O Q > r = O P .$ So $P$ beats every other point of $ℓ$ for nearness to $O$ (Fig. 10.4).

Now suppose $O P$ were not perpendicular to $ℓ .$ Draw $O M ⊥ ℓ$ with $M$ on $ℓ .$ As $O P$ is not perpendicular to the line, $M$ is a different point. In right triangle $O M P$ the hypotenuse $O P$ is longer than the side $O M,$ so $M$ would be nearer to $O$ than $P$ is. That is impossible, so $O P ⊥ ℓ .$ ∎

4/4 $O P$ is the shortest distance from $O$ to the tangent, so it meets the tangent at $90^{\circ} .$

Fig. 10.4Every point of the tangent except

P

is farther from

O

than

P

is.NCERT Fig. 10.5 · p. 147

A circle with centre O and a radius OP of 3 cm; a slanted tangent touches it at P. In the first step only the circle, the tangent and the radius are shown. In the second step a point Q on the tangent 4 cm from P is joined to O by a dashed line, and the label reads OQ = 5 cm, more than 3 cm. In the third step Q is 2.5 cm from P on the other side and OQ is about 3.9 cm, again more than 3 cm. In the fourth step a right-angle mark appears at P with the note that OP is the shortest distance, so there is a right angle at P.

SimulationTangent and radius explorer

The stage shows a circle, its centre O, a point P on the rim and the tangent at P. The radius OP is drawn with a right-angle mark at P. A second point Q sits on the tangent and is joined to O by a dashed line. You can drag P anywhere on the circle, and the tangent turns with it, and drag Q along the tangent. Readouts show OP, OQ, PQ and the angle OPQ, which stays 90 degrees, while OQ is always longer than the radius.

Fig. 10.5Drag

P

around the circle and

Q

along the tangent at

P .

Try this

Drag $Q$ towards $P .$ Can you ever make $O Q$ shorter than $O P$ ?
Move $P$ to three different places on the circle. Does $∠ O P Q$ ever change?
Read off $O P,$ $P Q$ and $O Q,$ and check that $O Q^{2} = O P^{2} + P Q^{2} .$

Because the angle at the point of contact is a right angle, every radius–tangent pair gives a right triangle, and Pythagoras finds missing lengths.

Example

Length of a tangent by Pythagoras

A circular pond has centre $O$ and radius 7 m. Asha stands at a point $Q,$ 25 m from $O .$ At a point $P,$ a straight path from $Q$ just grazes the pond. How far is it from $Q$ to $P$ ?

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 3

Q P

touches a circle with centre

O

P,

and

∠ P Q O = 35^{\circ} .

What is

∠ P O Q

35^{\circ}

45^{\circ}

55^{\circ}

145^{\circ}

Show answer

Answer (C) $55^{\circ}$

∠ O P Q = 90^{\circ}

(Theorem 10.1), so

∠ P O Q = 180^{\circ} - 90^{\circ} - 35^{\circ} = 55^{\circ} .

The answer

145^{\circ}

forgets the right angle at

P .

Practice · 4 questions

Practice 10.1: questions with answers

Short answerEasyNCERT Exercise 10.1, Q1
Riya says: “A circle has only two tangents, because I can fit my ruler against it on just two sides.” Is she right? How many tangents does a circle have?
exactly twoexactly fournoneone at each point, so infinitely many
Show solution
Riya is wrong. Every point of a circle has exactly one tangent (Two consequences), and a circle has infinitely many points, so it has infinitely many tangents. What Riya noticed is that, for one fixed direction of the ruler, only two tangents are parallel to it (Activity 10.2).
2
Quick answerEasyNCERT Exercise 10.1, Q2
Fill in the blanks.
1. A line that has exactly one point in common with a circle is called a
  Show answer
  Answer tangent
  One common point: a tangent.
2. A secant and a circle have points in common.
  Show answer
  Answer two
  A secant cuts the circle at two points.
3. Of all the lines parallel to one given line, at most can be tangents to a circle.
  Show answer
  Answer two
  Only the two lines at distance $r$ from the centre, one on each side, touch the circle.
4. A tangent touches a circle at $P .$ The point $P$ is called the
  Show answer
  Answer point of contact
  It is the point of contact.
Quick answerMediumNCERT Exercise 10.1, Q3
$O$ is the centre of a circle of radius 9 cm. A line through a point $Q$ touches the circle at $P,$ and $O Q = 41$ cm. The length $Q P$ is
$32$ cm $40$ cm $50$ cm $\sqrt{1762}$ cm
Show answer
Answer (B) $40$ cm
$∠ O P Q = 90^{\circ},$ so $Q P = \sqrt{41^{2} - 9^{2}} = \sqrt{1600} = 40$ cm. The values 32 and 50 subtract or add the lengths, and $\sqrt{1762}$ adds the squares.
4
DiagramMediumNCERT Exercise 10.1, Q4
Draw a circle of radius 3 cm with centre $O,$ and a line $m$ that is 5 cm from $O .$ Then draw (a) a line parallel to $m$ that is a tangent to the circle, and (b) a line parallel to $m$ that is a secant. How far is each of your lines from $O$ ?
Show solution
AnswerThe tangent is 3 cm from $O$ (there are two choices); a secant is less than 3 cm from $O,$ for example the line through $O .$
Draw the circle and the line $m .$ With a set square draw $O K ⊥ m,$ so $O K = 5$ cm.
$O K$ crosses the circle at a point $A,$ 3 cm from $O .$ Through $A$ draw the line perpendicular to $O A;$ it is parallel to $m .$ It is 3 cm from $O,$ so it touches the circle at $A :$ a tangent.
Through $O$ draw a line parallel to $m .$ It is 0 cm from $O,$ less than the radius, so it cuts the circle twice: a secant (in fact a diameter).
Any parallel line closer than 3 cm to $O$ is a secant. A second tangent lies on the far side, through the other end $B$ of the diameter along $O K .$

10.3 Number of Tangents from a Point on a Circle

Each point on a circle has exactly one tangent. Now turn the question around. Pick a point $P$ anywhere in the plane, perhaps inside the circle or far outside it. How many tangents to the circle pass through $P$ ? The answer depends only on where $P$ is.

Activity 10.3NCERT Activity 3 · p. 148

Inside, on or outside?

You’ll need

a coin or a bangle to trace
a ruler
a pencil

Predict first

From a point outside a circle, how many lines can you draw that just touch the circle?

exactly twoexactly onenoneas many as you like

Trace a circle. Mark a point $A$ inside it, a point $B$ on it and a point $C$ outside it.
Place the ruler's edge through $A$ and turn it slowly about $A .$ Look for a position in which the edge only touches the circle.
Do the same for $B$ and for $C .$ Draw every touching line you find, and fill the table.

Table 10.4

Point	Where it is	Tangents
$A$	inside
$B$	on the circle
$C$	outside

Your entries are saved automatically.

3/3From an outside point there are exactly two tangents, touching at $A$ and $B .$

Fig. 10.6The number of tangents through

P

depends on where

P

is.NCERT Fig. 10.6 · p. 148

A circle with centre O and a point P on a horizontal line through O. In the first step P is inside the circle; three lines through P are drawn and each crosses the circle at two points, and the panel reads Inside, 0 tangents. In the second step P has moved onto the circle and a single vertical tangent passes through it; the panel reads On, 1 tangent. In the third step P is outside the circle and two tangents from P touch the circle at A and B; the panel reads Outside, 2 tangents.

$P$ inside the circle: no tangent passes through $P .$
$P$ on the circle: exactly one tangent passes through $P .$
$P$ outside the circle: exactly two tangents pass through $P .$

Think about it

Why none from inside, and exactly two from outside?

From inside. Any line through $P$ is at most $O P$ away from $O,$ and $O P$ is less than the radius. By (10.1) the line cuts a chord of positive length, so it is a secant.

From outside. Suppose a tangent from $P$ touches the circle at $A .$ By Theorem 10.1, the angle $O A P$ is a right angle. In Class 9 you saw that an angle in a semicircle is a right angle. The converse is also true: a right angle that looks at the segment $O P$ has its corner on the circle drawn on $O P$ as diameter. So every point of contact lies on two circles at once: the given circle and the circle on $O P$ as diameter (Fig. 10.7).

The circle on $O P$ passes through $O,$ inside the given circle, and through $P,$ outside it, so it crosses the given circle. Two different circles share at most two points, so they cross at exactly two points, $A$ and $B .$ Hence there are exactly two tangents.

3/3It crosses the given circle at $A$ and $B,$ where $∠ O A P = ∠ O B P = 90^{\circ} :$ the two points of contact.

Fig. 10.7The points of contact lie on the circle with diameter

O P .

A circle with centre O and an outside point P joined to O by a dashed segment. In the second step a dashed green circle is drawn with OP as its diameter and its centre M at the midpoint of OP. In the third step the two points A and B where the circles cross are marked; the radii OA and OB and the tangents PA and PB are drawn, with right-angle marks at A and B.

Definition

Length of a tangent

For a point $P$ outside a circle, the length of the tangent from $P$ is the distance $P A$ from $P$ to the point of contact $A .$

Look again at the two tangents from $P$ in Fig. 10.6. The segments $P A$ and $P B$ look exactly the same length, and a ruler agrees. A proof shows that this happens for every outside point.

Theorem 10.2Equal tangents

The two tangents drawn to a circle from a point outside it have equal lengths.

Given: $P$ is a point outside a circle whose centre is $O,$ and the two tangents from $P$ touch the circle at $A$ and $B .$

To prove: $P A = P B .$

Construction: join $O A,$ $O B$ and $O P$ (Fig. 10.8).

Proof: By Theorem 10.1, each radius meets its tangent at a right angle, so $∠ O A P = 90^{\circ}$ and $∠ O B P = 90^{\circ} .$ Now compare the right triangles $O A P$ and $O B P :$

their hypotenuse $O P$ is common;
$O A = O B,$ since both are radii.

So $△ O A P ≅ △ O B P$ by the RHS rule, and the matching sides $P A$ and $P B$ are equal (CPCT). ∎

4/4By RHS the triangles are congruent, so $P A = P B$ and $P O$ bisects $∠ A P B .$

Fig. 10.8Two right triangles with a common hypotenuse and equal radii are congruent.NCERT Fig. 10.7 · p. 149

A circle with centre O at the upper left and a point P at the lower right. In the first step the two tangents PA and PB are drawn. In the second step the radii OA and OB are added with right-angle marks at A and B, and P is joined to O by a dashed line. In the third step triangle OAP is shaded blue and triangle OBP green. In the fourth step PA and PB carry equal tick marks, the two angles at P between each tangent and PO carry arcs, and the label reads PA = PB.

The Pythagoras form gives a formula for the tangent length from any outside point $P :$

\begin{aligned} P A & = P B \\ = \sqrt{O P^{2} - r^{2}} \end{aligned}

(10.2)

where

$O P$: the distance of the point from the centre
$r$: the radius

InteractiveTangent lab

A circle of radius 3 cm with centre O and three modes. In Point mode you drag a point P, or set its distance and direction with sliders: inside the circle three sample lines through P all cut the circle twice and the panel says no tangent; on the circle one tangent appears at right angles to OP; outside, the two tangents PA and PB appear with radii and right-angle marks, and a table shows OP, PA and PB (always equal), the value of the square root of OP squared minus r squared, and that angle APB plus angle AOB is 180 degrees. In Turn mode a slider turns a line about a point P of the circle; the chord PQ shrinks and becomes zero when the line is at 90 degrees to OP. In Slide mode a horizontal line moves up and down; its chord is 2 times the square root of 9 minus d squared, and it becomes a tangent at 3 cm from O.

Fig. 10.9Drag

P

inside, onto and outside the circle, or switch to the Turn and Slide modes.

The three examples below use the two theorems together, the way board questions do.

3/3 $O A = 65$ and $O P = 25,$ so $A P = 60$ and $A B = 120$ (in cm).

Fig. 10.10The touching point

P

is the midpoint of the chord

A B .

NCERT Fig. 10.8 · p. 149

Two circles with the same centre O, with a ring of dots around them like a rangoli. A horizontal chord AB of the larger circle touches the smaller circle at P, directly above O. In the second step the radius OP is drawn with a right-angle mark at P, and AP and PB carry equal tick marks. In the third step OA is drawn dashed and the lengths are labelled: OP 25, OA 65 and AP 60, in centimetres.

Example 10.1NCERT Example 1 · p. 149

A rangoli line

For a rangoli, Meera draws two circles with the same centre $O$ (concentric circles), of radii 65 cm and 25 cm. She lays a straight line of white powder $A B,$ a chord of the bigger circle, which touches the smaller circle at $P$ and nowhere else. (a) Prove that $P$ is the midpoint of $A B .$ (b) Find the length of $A B .$

Try it yourself first. Then reveal the solution one step at a time.

3/3 $∠ O A P = 90^{\circ},$ so $∠ O A B = 90^{\circ} - 65^{\circ} = 25^{\circ} .$

Fig. 10.11

∠ O A B

is half of

∠ A P B .

NCERT Fig. 10.9 · p. 150

A circle with centre O near the top and a point P below it; two tangents from P touch the circle at A and B, and the angle at P is marked 50 degrees. In the second step the chord AB is drawn, the tangents carry equal tick marks, and the base angles of triangle PAB are marked 65 degrees each. In the third step the radius OA is drawn with a right-angle mark at A, and the angle between OA and AB is marked 25 degrees.

Example 10.2NCERT Example 2 · p. 150

Angles made by two tangents

Tangents from an outside point $P$ touch a circle (centre $O$ ) at $A$ and $B,$ and $∠ A P B = 50^{\circ} .$ (a) Find the angles $P A B$ and $O A B$ in degrees. (b) Prove that $∠ O A B = \frac{1}{2} ∠ A P B$ for any two tangents from an outside point.

Try it yourself first. Then reveal the solution one step at a time.

3/3 $∠ O A T = 90^{\circ},$ and similar triangles give $T A = 7.5$ m.

Fig. 10.12The fence

T A,

the radius

O A

and the line

O T

form a right triangle.NCERT Fig. 10.10 · p. 150

A green circular garden with centre O. A straight path AB near the top is labelled 12 m, and two fences along the tangents at A and B meet at the gate T above the garden. In the second step T is joined to O by a dashed line crossing AB at M with a right angle; AM is labelled 6, OM 8 and the radius OA 10. In the third step a right-angle mark appears at A between the radius and the fence, and TA is labelled 7.5, all in metres.

Example 10.3NCERT Example 3 · p. 150

Fences along two tangents

A circular garden has centre $O$ and radius 10 m. A straight path $A B,$ 12 m long, joins two points on its edge. Fences are built along the tangents at $A$ and $B,$ and they meet at a gate $T .$ Find the length $T A$ of each fence.

Try it yourself first. Then reveal the solution one step at a time.

3/3Each side is a sum of two letters, and the perimeter is $2 (x + y + z) .$

Fig. 10.13Equal tangents from each corner split every side into two labelled parts.

A triangle ABC whose three sides touch a circle drawn inside it. In the second step the two tangent segments from A are coloured blue and labelled x, the two from B green and labelled y, and the two from C orange and labelled z. In the third step a panel on the right adds them up: AB = x + y, BC = y + z, CA = z + x, so the perimeter is 2x + 2y + 2z.

Quick check

1 of 4

Tangents from

P

to a circle of radius 5 cm are 12 cm long. How far is

P

from the centre?

Show answer

Answer 13 cm

O P = \sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13

cm.

10.4 Summary

1/4Three facts carry the whole chapter.

Fig. 10.14The three big ideas of the chapter.

Three panels side by side. The first shows a circle with three horizontal lines: a dashed one that misses it, a blue tangent touching it at one point and a purple secant crossing it at two points, captioned 0, 1 or 2 points. The second shows a radius meeting a tangent with a right-angle mark, captioned right angle at contact. The third shows two tangents from an outside point P with equal tick marks, captioned equal tangents. The first step shows all three panels; each later step brightens one panel and dims the other two.

Recap

Circles: key points and summary

A line and a circle share no point (non-intersecting line), one point (tangent) or two points (secant), never three (Section 10.1).
If $d$ is the distance of a line from the centre, the line misses the circle when $d > r,$ touches it when $d = r$ and cuts it when $d < r$ ((10.1)).
A tangent is the limiting position of a secant whose two meeting points come together (Activity 10.1).
At most two tangents are parallel to a given line (Activity 10.2).
The tangent at a point is perpendicular to the radius drawn to the point of contact (Theorem 10.1).
Each point of a circle has exactly one tangent; the line through the centre and the point of contact is the normal (Two consequences).
Through a point inside a circle pass no tangents, through a point on it one, and through a point outside it exactly two (Section 10.3).
The two tangents from an outside point are equal (Theorem 10.2).
The tangent length from $P$ is $\sqrt{O P^{2} - r^{2}}$ ((10.2)).
$O P$ bisects the angle between the tangents and is the perpendicular bisector of the chord $A B$ (More from the same two triangles).
The angle between two tangents and the angle at the centre between the radii to the points of contact add up to 180° (Practice 10.2, Q10).
Around a circle, each corner of a polygon gives a pair of equal tangents (Count the tangent pairs).

Flash cards · 6

Key terms

1 / 6

Board exam practice

Find your NCERT exercise question

Our practice questions are written for ncert.ai and follow the NCERT exercises. Find the one that matches the question in your textbook; its answer is with it.

In your NCERT textbook	Practice question here
NCERT Exercise 10.1, Q1	Practice 10.1 · Q1
NCERT Exercise 10.1, Q2	Practice 10.1 · Q2
NCERT Exercise 10.1, Q3	Practice 10.1 · Q3
NCERT Exercise 10.1, Q4	Practice 10.1 · Q4
NCERT Exercise 10.2, Q1	Practice 10.2 · Q1
NCERT Exercise 10.2, Q2	Practice 10.2 · Q2
NCERT Exercise 10.2, Q3	Practice 10.2 · Q3
NCERT Exercise 10.2, Q4	Practice 10.2 · Q4
NCERT Exercise 10.2, Q5	Practice 10.2 · Q5
NCERT Exercise 10.2, Q6	Practice 10.2 · Q6
NCERT Exercise 10.2, Q7	Practice 10.2 · Q7
NCERT Exercise 10.2, Q8	Practice 10.2 · Q8
NCERT Exercise 10.2, Q9	Practice 10.2 · Q9
NCERT Exercise 10.2, Q10	Practice 10.2 · Q10
NCERT Exercise 10.2, Q11	Practice 10.2 · Q11
NCERT Exercise 10.2, Q12	Practice 10.2 · Q12
NCERT Exercise 10.2, Q13	Practice 10.2 · Q13

Quick check: MCQs on Circles

10.1 Introduction3 questions
10.2 Tangent to a Circle3 questions
10.3 Number of Tangents from a Point on a Circle4 questions

Glossary · 9

Key terms (English–Hindi)

circleवृत्तvritt: All the points of a plane that are one fixed distance (the radius) from a fixed point (the centre).First seen in the chapter →
circumscribeपरिगतparigat: A polygon circumscribes a circle when every side of the polygon is a tangent to the circle.First seen in the chapter →
concentric circlesसंकेंद्रीय वृत्तsankendriya vritt: Circles that have the same centre.First seen in the chapter →

length of a tangentस्पर्श रेखा की लंबाईsparsh rekha ki lambai: For a point $P$ outside a circle, the distance from $P$ to the point of contact of a tangent drawn from $P .$

non-intersecting lineअप्रतिच्छेदी रेखाapratichchhedi rekha: A line that has no point in common with a given circle.First seen in the chapter →
normalअभिलंबabhilamb: The line through the centre of a circle and a point of contact; it is perpendicular to the tangent there.First seen in the chapter →

point of contactस्पर्श बिंदुsparsh bindu: The single point that a tangent shares with a circle.

secantछेदक रेखाchhedak rekha: A line that meets a circle at two points.First seen in the chapter →

tangentस्पर्श रेखाsparsh rekha: A line that meets a circle at exactly one point, its point of contact.First seen in the chapter →

Circles: Class 10 Maths · Chapter 10

What you will learn in Circles

10.1 Introduction

10.2 Tangent to a Circle

Practice 10.1: questions with answers

10.3 Number of Tangents from a Point on a Circle

Practice 10.2: questions with answers

10.4 Summary

Circles: key points and summary

Board exam practice

Practice questions with answers (NCERT “Board exam practice”)

Find your NCERT exercise question

Quick check: MCQs on Circles

Key terms (English–Hindi)

You reached the end