Class 10 Maths - Circles

Draw two tangents from a point $5$ cm away from the centre of a circle of radius $3$ cm.

Given, a circle of radius

$3$ cm whose Centre is

$O$ and a point P

$, 5$ cm away from its centre.

STEPS OF CONSTRUCTION:

$1.$ Draw circle with

$O$ as centre and radius

$= 3 cm.$ Take a point

$P$ such that

$OP= 5 cm.$

$2.$ Draw the bisector of

$OP$ which intersect

$OP$ at

$M.$

$3.$ Taking

$M$ as centre and

$MO$ as radius, draw a dotted circle. Let this circle cuts the given circle

$A$ and

$B.$

$4.$ Join

$PA$ &

$PB$ .

$5.$ Thus ,

$PA$ &

$PB$ are the required tangents. By measurement using scale

$PA= PB = 4 cm.$

1316380_1068946_ans_8b7409dc3d2e4a1383c37f9350ce6d1b.png

Maximum number of tangents that can be drawn from an external point to a circle is ______ .

The tangent of a circle is a special case of the secant when the two end points of the corresponding chord coincide.

From an external point

$P$ , only two tangents (

$PA$ and

$PB$ ) can be drawn.

Also, both the drawn tangents are equal in length.

Hence, maximum

$2$ tangents can be drawn from an external point to a circle.

How many tangents can be drawn on the circle of radius $5\ \text{cm}$ from a point lying outside the circle at distance $9\ \text{cm}$ from the center.

Number of tangents that can be drawn to a circle from a point lying outside the circle is two.

Since, distance between the center and the point (

$9\ \text{cm}$ ) is greater than the radius (

$5\ \text{cm}$ ), the point lies outside the circle. Hence, two tangents can be drawn on the circle.

i) A circle can have _ parallel tangents.
ii) The point common to the tangent and the circle is called ___.

i) A circle can have two parallel tangents.

ii) The point common to the tangent and the circle is called the point of contact.

What is the locus of the centre of a wheel of a bicycle going straight along a level road.

The locus of the centre of a wheel, which is going straight along a level road will be a straight line parallel to the road at a distance equal to the radius of the wheel.
1796986_1832813_ans_c626dd938aa74db7b3ade64d590f75d1.png

1796986_1832813_ans_c626dd938aa74db7b3ade64d590f75d1.png

Find the equation of the tangent to the circle $\displaystyle x^{2}+y^{2}-30x+6y+109=0$ at (4, -1)

Equation of tangent is

$\displaystyle 4x+(-y)-30\left ( \frac{x+4}{2} \right )+6\left ( \frac{y+(-1)}{2} \right )+109=0$
or

$4x - y - 15x - 60 + 37 - 3 + 109 = 0 or -11x + 2y + 46 = 0$
or

$11x - 2y - 46 = 0$
Hence the required equation of the tangent is

$11x - 2y - 46 = 0$

The number of tangents that can be drawn to a circle at a given point on it is

Consider given the two dice,

If the point is on the circle then number of tangent $= 1$

If the point is outside the circle then number of tangent $=2$

If the point is in the circle then number of tangent $=$ none

Hence, this is the answer.

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD in cm.

From the graph, we find some conclusion-

$PA=PB=10$ cm

$CE=CA$

$DE=DB$

Perimeter of

$\Delta PCD$ is,

$=PC+PD+CD$

$\Rightarrow PC+PD+CE+ED$

$\Rightarrow PC+PD+CA+DB$

$\Rightarrow (PC+CA)(PD+DB)$

$\Rightarrow PA+PB$

$\Rightarrow 10+10$

$\Rightarrow 20$ cm.

1458599_427348_ans_7797fef64bd241dcb1f1f40ac46b37a1.png

In the ddjoining figure $O$ is the centre of the circle. Forom point $R\ seg\ RM$ and $seg\ RN$ are tengent segment touching the circle at $M$ and $N$ . If $(OR)=10\ cm$ and radius of the circle $=5\ cm$ , then
What is the measure os $\angle MRN$ ?

We have

$OM=5$ cm ad

$OR=10$ cm

In rightangled

$\triangle{OMR},$

${OR}^{2}={OM}^{2}+{MR}^{2}$

$\Rightarrow MR=\sqrt{{OR}^{2}-{OM}^{2}}=\sqrt{{10}^{2}-{5}^{2}}=\sqrt{100-25}=\sqrt{75}=5\sqrt{3}$ cm

Tangent segments drawn from an external point to a circle are congruent.

$\therefore MR=NR=5\sqrt{3}$ cm

In right angled

$\triangle{OMR}$

$\tan{\angle{MRO}}=\dfrac{OM}{MR}$

$=\dfrac{5}{5\sqrt{3}}=\dfrac{1}{\sqrt{3}}$

$\Rightarrow \tan{\angle{MRO}}=\tan{{30}^{\circ}}$

Thus, the measure of

$\angle{MRO}$ is

${30}^{\circ}$

In right angled

$\triangle{ONR}$

$\tan{\angle{NRO}}=\dfrac{ON}{NR}$

$=\dfrac{5}{5\sqrt{3}}=\dfrac{1}{\sqrt{3}}$

$\Rightarrow \tan{\angle{NRO}}=\tan{{30}^{\circ}}$

Thus, the measure of

$\angle{NRO}$ is

${30}^{\circ}$

$\therefore \angle{MRN}=\angle{MRO}+\angle{NRO}={30}^{\circ}+{30}^{\circ}={60}^{\circ}$

Thus, the measure of

$\angle{MRN}$ is

${60}^{\circ}$

If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that $\angle DBC = {120^0}$ prove that BC+BD=BO, i.e, BO=2BC.

Sol. Given : A circle with centre O.

Tangents BC and BD are drawn from an external point B such that

$\angle DBC = 120^{\circ}$

To prove :

$BC + BD = BO,$ i.e.,

$BO = 2BC$

Construction : join

$OB, OC$ and

$OD$ .

Proof : In

$\Delta OBC$ and

$\Delta OBD$ , we have

$OB = OB$ [common]

$OC = OD$ [Raddi of same circle]

$BC = BD$ [Tangents from an external point are equal in length ] __(i)

$\therefore \Delta OBC \cong \Delta OBD$ [By SSS criterion of congruence]

$\Rightarrow \angle OBC = \angle OBD (CPCT)$

$\therefore \angle OBC = \dfrac{1}{2} \angle DBC = \dfrac{1}{2} \times 120^{\circ}$ [

$\because \angle CBD = 120^{\circ}$ given]

$\Rightarrow \angle OBC = 60^{\circ}$

OC and BC are radius and tangent respectively at contact point C.

So,

$\angle OCB = 90^{\circ}$

Now, in right angle

$\Delta OBC = 60^{\circ}$

$\therefore cos 60^{\circ} = \dfrac{BC}{BO}$

$\Rightarrow \dfrac{1}{2} = \dfrac{BC}{BO}$

$\Rightarrow OB = 2 BC$

Hence, proved (ii) part,

$\Rightarrow OB = BC + BC$

$\Rightarrow OB = BC + BD$ [

$\because BC =BD$ from (i)]

Hence, proved.

1791821_1458800_ans_255966551fee44788fd0f37fae1b9ede.png

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60 .

${\textbf{Step - 1: Finding various angles}}{\text{.}}$

${\text{As we have to make to angle between the tangents equal to 60}}^\circ$

${\text{and the tangents subtend an angle of 90}}^\circ {\text{ at its point of contact}}{\text{.}}$

${\text{We can write, that the sum of all the angles of a quadrilateral is equal to 360}}^\circ {\text{,}}$

${\text{So for the quadrilateral we have the three angles equal to 60}}^\circ ,90^\circ ,{\text{and 90}}^\circ$

${\text{Therefore,the angle subtended by the pair of tangents at the centre will be }}$

$360^\circ - 90^\circ - 90^\circ - 60^\circ = 120^\circ$

${\textbf{Step - 2: Drawing the figure}}{\text{.}}$

${\textbf{Hence, the pair of the tangents can be drawn as above}}{\text{.}}$

Two tangents PA and PB are drawn to a circle with centre O, such that $\angle APB=120^\circ$ Prove that $OP=AP+BP=2AP$

Given: O is the centre of the circle. PA and PB are tangents drawn to a circle and ∠APB = 120°.

To prove: OP = 2AP

Proof:

In ΔOAP and ΔOBP,

OP = OP (Common)

∠OAP = ∠OBP (90°) (Radius is perpendicular to the tangent at the point of contact)

OA = OB (Radius of the circle)

∴ ΔOAP is congruent to ΔOBP (RHS criterion)

∠OPA = ∠OPB = 120°/2 = 60° (CPCT)

In ΔOAP,

cos∠OPA = cos 60° = AP/OP

Therefore, 1/2 =AP/OP

Thus, OP = 2AP

Hence, proved.

A circle touches the side $BC$ of $\triangle\ ABC$ at $P$ and touches $AB$ and $AC$ produced at $Q$ and $R$ respectively. Prove that $AQ=\dfrac{1}{2}$ (perimeter of $\triangle\ ABC$ ).

Given:A circle touching the side

$BC$ of

$\triangle{ABC}$ at

$P$ and

$AB$ ,

$AC$ produced at

$Q$ and

$R$ respectively.

Proof: Lengths of tangents drawn from an external point to a circle are equal.

$\Rightarrow AQ = AR, BQ = BP, CP = CR$ .

Perimeter of

$\triangle{ABC}=AB + BC + CA$

$= AB +\left(BP + PC\right)+\left(AR – CR\right)$

$=\left(AB + BQ\right)+\left(PC\right)+\left(AQ – PC\right)$ since

$\left[AQ = AR, BQ = BP, CP = CR\right]$

$= AQ + AQ$

$= 2AQ$

$\Rightarrow AQ =\dfrac{1}{2}$ (Perimeter of

$\triangle{ABC}$ )

$\therefore AQ$ is the half of the perimeter of

$\triangle{ABC}$

Write the name of the common point of the tangent to a circle and the circle.

The common point of the tangent to a circle and the circle is called the point of contact.

Which term will you use for a line that intersects a circle at two distinct points?

The term used for a line that intersects a circle at two distinct points is secant. When a line touches a circle at only one point it is known as tangent.

How many tangents can a circle have?

${\textbf{Step1: Find the number of tangents to the circle from the given points. }}$

${\text{ Circle is the locus of the points equidistant from given points, which is the center of the circle }}$

${\text{and tangent is the line that intersects a circle at one point only.}}$

${\text{ As circle has infinite points so there will be infinite tangents can be drawn on these points}}$

${\text{ which touches at only one point.}}$

${\textbf{Therefore, a circle can have an infinite number of tangents .}}$

$DA$ and $DB$ are the triangle drwan from D, to the circle with centre $O$ , at A and B respectively $\angle ADO = 35^\circ$ find $\angle AOB$

We are given,

Circle of centre

$O$ is having radius

$OA$ and

$OB$ (r)

Now, In

$\Delta OAD$ &

$\Delta OBD$ is equal as two sides are equal and one angle is equal

$(OA = OB = r)$ and

$OD$ also

$m\angle OAD = m \angle OBD = 90^o$

Now, given that

$m\angle OBD = 90^o$

Now, given that

$m \angle ADO = 35^o$

then

$m\angle AOD = ?$

$\Delta AOD, m \angle AOD + m \angle ADO + m \angle OAD = 180^o$

$m\angle AOD = 180 - m \angle OAD - m \angle ADO$

$m \angle AOD = 180 - 90 - 35 = 55^o$

So,

$m\angle AOB = m \angle AOD + m \angle DOB$

$= 55^o + 55^o$

$m \angle AOB = 110^o$

1493219_1529218_ans_eb99b5c0f81b4e1fa8d5799c21dbd6a0.png

Pair of tangents are drawn from point (1,3) to circle ${ x }^{ 2 }+{ y }^{ 2 }-3x-x+3y-1=0$ Find line joining their point of contact

Equation of the circle is given as,

$x^2+y^2-3x-x+3y-1=0$ ……………..

$(1)$

$\Rightarrow x^2+y^2-4x+3y-1=0$

$\Rightarrow x^2+y^2+2(-2)x+2.\left(\dfrac{3}{2}\right)y+(-1)=0$

$\therefore g=-2$ ,

$f=\dfrac{3}{2}$ and

$c=-1$

Equation of the chord of contact of circle

$(1)$ is given as

$x_1x+y_1y+g(x_1+x)+f(y_1+y)+c=0$

$\Rightarrow x_1x+y_1y+(-2)(x_1+x)+\dfrac{3}{2}(y_1+y)-1=0$

Now, equation of the chord of contact of circle

$(1)$ wrt to

$(1, 3)$

$1x+3y+(-2)(1+x)+\dfrac{3}{2}(3+y)-1=0$

$2x+6y-4(1+x)+3(3+y)-2=0$

__________________________________

$2$

$\Rightarrow 2x+6y-4-4x+9+3y-2=0$

$\Rightarrow -2x+9y+3=0$

$\Rightarrow 2x-9y-3=0$

$\Rightarrow 2x-9y=3$ .

1215675_1300969_ans_2cb011034e1b48248997ae8fa3fcdb78.JPG

How many tangents can be drawn at a point on a circle ?

Only one tangent can be drawn from a point on the circle
Every other line intersect the circle at two points which won't be a tangent
1795328_1815673_ans_c6f48f0fac564444a03d703a9f5a5211.png

1795328_1815673_ans_c6f48f0fac564444a03d703a9f5a5211.png

Draw two concentric circles of radii $3 \,cm$ and $5 \,cm.$ Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

Steps of construction:

(i) Draw two concentric circles

$C_1, C_2$ of radii

$3 \,cm$ and

$5 \,cm$ respectively taking 'O' as centre.

(ii) Draw perpendicular bisector AB of OT. T is any point on

$C_2.$

(iii) Draw circle

$C_3$ taking radius

$TM = OM$ and

$M$ as centre.

(iv) Circle

$C_3$ intersect the circle

$C_1$ at P and Q. Join TP and TQ. These are the required tangents.

$TP = TQ = 4.1 \,cm$ by measuring.

Mathematically length of tangent: Join OP. OP and TP are radius and tangent respectively at contact point P. So,

$\angle TPO = 90^\circ.$

By Pythagoras theorem in

$\Delta TPO,$

$PT^2 = OT^2 - OP^2 = 5^2 - 3^2 = 25 - 9 = 16$

$\Rightarrow PT = 4 \,cm$

Difference in measurement and by mathematical calculation

$PT = 4.1 \,cm - 4 \,cm = 0.1 \,cm.$

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre 0 and radius r at P. Q and R respectively.
Prove That :
(i) AB+CQ=AC+BQ
(ii) Area $(\Delta ABQ=\frac{1}{2}$ (Perimeter of $(\Delta ABC)\times r$

2173000_1582989_ans_71369e9dcad64cb3a2802c108f504b34.jpeg

In the given figure, tangents $PQ$ and $PR$ are drawn from an external point p to a circle with center $O$ , such that $\angle RPQ=30^{\circ}$ . A chord $RS$ is drawn parallel to the tangent $PQ$ . Find $\angle RQS.$

1570510_971375_ans_99a8ec3d198147ca917a817416ee48b4.jpg

In the given figure, O is the centre of a circle of radius 5 cm, T is a point such that OT= 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

$OP = OQ = 5 cm$

$OT = 13 cm$

$OP$ and

$PT$ are radius and tangent respectively at contact point P.

$\therefore \angle OPT = 90^{\circ}$

So, by pythagoras theorem, in right angled

$\Delta OPT$ ,

$PT^2 = OT^2 - OP^2 = 13^2 - 5^2 = 169 - 25 = 144$

$\Rightarrow PT = 12 cm$

$AP$ and

$AE$ are two tangents from an external point A to a circle.

$\therefore AP = AE$

$AEB$ is tangent and

$OE$ is radius at contact point E.

So,

$AB \bot OT$ ___(i)

So, by Pythagoras theorem, in right angled.

$\Delta AET$ .

$AE^2 = AT^2 - ET^2$

$\Rightarrow AE^2 = (PT - PA)^2 - [TO - OE]^2$

$=(12 - AE)^2 - (13 - 5)^2$

$\Rightarrow AE^2 = (12)^2 + (AE)^2 - 2 (12) (AE) - (8)^2$

$\Rightarrow AE^2 - AE^2 + 24 AE = 144 - 64$

$\Rightarrow 24 AE = 80$

$\Rightarrow AE = \dfrac{80}{24} cm$

$\Rightarrow AE = \dfrac{10}{3} cm$

REF. IMAGE

$\Delta TPO$ and

$\Delta TQO$ ,

$OT = OT$ [common]

$PT = QT$ [Tangents from T]

$OP = OQ$ [Radii of same circle]

$\therefore \Delta TPO \cong \Delta TQO$ [By SSS criterion of congruence]

$\Rightarrow \angle 1 = \angle 2$ ___(ii) [CPCT]

$\Delta ETA$ and

$\Delta ETB$ ,

$ET = ET$ [Common]

$\angle TEA = \angle TEB = 90^{\circ}$ [From (i)]

$\angle 1 = \angle 2$ [CPCT] [From (ii)]

$\therefore \Delta ETA \cong \Delta ETB$ [By ASA criterion of congruence]

$\Rightarrow AE = BE$ [CPCT]

$\Rightarrow AB = 2 AE = 2 \times \dfrac{10}{3}$

$\Rightarrow AB = \dfrac{20}{3} cm$

Hence, the required length is

$\dfrac{20}{3} cm$ .

1804487_1516190_ans_a7fea1e0002d4770b10122e7a1979cb7.png

Circles - Class 10 Maths - Extra Questions

Draw two tangents from a point $5$ cm away from the centre of a circle of radius $3$ cm.

Maximum number of tangents that can be drawn from an external point to a circle is ______ .

How many tangents can be drawn on the circle of radius $5\ \text{cm}$ from a point lying outside the circle at distance $9\ \text{cm}$ from the center.

i) A circle can have _ parallel tangents.
ii) The point common to the tangent and the circle is called ___.

What is the locus of the centre of a wheel of a bicycle going straight along a level road.

Find the equation of the tangent to the circle $\displaystyle x^{2}+y^{2}-30x+6y+109=0$ at (4, -1)

The number of tangents that can be drawn to a circle at a given point on it is

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD in cm.

In the ddjoining figure $O$ is the centre of the circle. Forom point $R\ seg\ RM$ and $seg\ RN$ are tengent segment touching the circle at $M$ and $N$ . If $(OR)=10\ cm$ and radius of the circle $=5\ cm$ , then
What is the measure os $\angle MRN$ ?

If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that $\angle DBC = {120^0}$ prove that BC+BD=BO, i.e, BO=2BC.

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60 .

Two tangents PA and PB are drawn to a circle with centre O, such that $\angle APB=120^\circ$ Prove that $OP=AP+BP=2AP$

A circle touches the side $BC$ of $\triangle\ ABC$ at $P$ and touches $AB$ and $AC$ produced at $Q$ and $R$ respectively. Prove that $AQ=\dfrac{1}{2}$ (perimeter of $\triangle\ ABC$ ).

Write the name of the common point of the tangent to a circle and the circle.

Which term will you use for a line that intersects a circle at two distinct points?

How many tangents can a circle have?

$DA$ and $DB$ are the triangle drwan from D, to the circle with centre $O$ , at A and B respectively $\angle ADO = 35^\circ$ find $\angle AOB$

Pair of tangents are drawn from point (1,3) to circle ${ x }^{ 2 }+{ y }^{ 2 }-3x-x+3y-1=0$ Find line joining their point of contact

How many tangents can be drawn at a point on a circle ?

Draw two concentric circles of radii $3 \,cm$ and $5 \,cm.$ Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre 0 and radius r at P. Q and R respectively.
Prove That :
(i) AB+CQ=AC+BQ
(ii) Area $(\Delta ABQ=\frac{1}{2}$ (Perimeter of $(\Delta ABC)\times r$

In the given figure, tangents $PQ$ and $PR$ are drawn from an external point p to a circle with center $O$ , such that $\angle RPQ=30^{\circ}$ . A chord $RS$ is drawn parallel to the tangent $PQ$ . Find $\angle RQS.$

In the given figure, O is the centre of a circle of radius 5 cm, T is a point such that OT= 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

Class 10 Maths Extra Questions

Circles - Class 10 Maths - Extra Questions

Draw two tangents from a point 55 cm away from the centre of a circle of radius 33 cm.

Maximum number of tangents that can be drawn from an external point to a circle is ______ .

How many tangents can be drawn on the circle of radius 5 cm5\ \text{cm} from a point lying outside the circle at distance 9 cm9\ \text{cm} from the center.

i) A circle can have ___ parallel tangents.ii) The point common to the tangent and the circle is called _____.

What is the locus of the centre of a wheel of a bicycle going straight along a level road.

Find the equation of the tangent to the circle x2+y2−30x+6y+109=0\displaystyle x^{2}+y^{2}-30x+6y+109=0 at (4, -1)

The number of tangents that can be drawn to a circle at a given point on it is

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD in cm.

In the ddjoining figure OO is the centre of the circle. Forom point R seg RMR\ seg\ RM and seg RNseg\ RN are tengent segment touching the circle at MM and NN. If (OR)=10 cm(OR)=10\ cm and radius of the circle =5 cm=5\ cm, thenWhat is the measure os ∠MRN\angle MRN?

If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that \angle DBC = {120^0}\angle DBC = {120^0} prove that BC+BD=BO, i.e, BO=2BC.

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60 .

Two tangents PA and PB are drawn to a circle with centre O, such that \angle APB=120^\circ\angle APB=120^\circ Prove that OP=AP+BP=2APOP=AP+BP=2AP

A circle touches the side BCBC of \triangle\ ABC\triangle\ ABC at PP and touches ABAB and ACAC produced at QQ and RR respectively. Prove that AQ=\dfrac{1}{2}AQ=\dfrac{1}{2} (perimeter of \triangle\ ABC\triangle\ ABC).

Write the name of the common point of the tangent to a circle and the circle.

Which term will you use for a line that intersects a circle at two distinct points?

How many tangents can a circle have?

DADA and DBDB are the triangle drwan from D, to the circle with centre OO, at A and B respectively \angle ADO = 35^\circ\angle ADO = 35^\circ find \angle AOB\angle AOB

Pair of tangents are drawn from point (1,3) to circle { x }^{ 2 }+{ y }^{ 2 }-3x-x+3y-1=0{ x }^{ 2 }+{ y }^{ 2 }-3x-x+3y-1=0 Find line joining their point of contact

How many tangents can be drawn at a point on a circle ?

Draw two concentric circles of radii 3 \,cm3 \,cm and 5 \,cm.5 \,cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre 0 and radius r at P. Q and R respectively.Prove That : (i) AB+CQ=AC+BQ(ii) Area (\Delta ABQ=\frac{1}{2}(\Delta ABQ=\frac{1}{2} (Perimeter of (\Delta ABC)\times r(\Delta ABC)\times r

In the given figure, tangents PQPQ and PRPR are drawn from an external point p to a circle with center OO, such that \angle RPQ=30^{\circ}\angle RPQ=30^{\circ}. A chord RSRS is drawn parallel to the tangent PQPQ. Find \angle RQS.\angle RQS.

In the given figure, O is the centre of a circle of radius 5 cm, T is a point such that OT= 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.

Class 10 Maths Extra Questions

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Draw two tangents from a point $5$ cm away from the centre of a circle of radius $3$ cm.

How many tangents can be drawn on the circle of radius $5\ \text{cm}$ from a point lying outside the circle at distance $9\ \text{cm}$ from the center.

i) A circle can have _ parallel tangents.
ii) The point common to the tangent and the circle is called ___.

Find the equation of the tangent to the circle $\displaystyle x^{2}+y^{2}-30x+6y+109=0$ at (4, -1)

In the ddjoining figure $O$ is the centre of the circle. Forom point $R\ seg\ RM$ and $seg\ RN$ are tengent segment touching the circle at $M$ and $N$ . If $(OR)=10\ cm$ and radius of the circle $=5\ cm$ , then
What is the measure os $\angle MRN$ ?

If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that $\angle DBC = {120^0}$ prove that BC+BD=BO, i.e, BO=2BC.

Two tangents PA and PB are drawn to a circle with centre O, such that $\angle APB=120^\circ$ Prove that $OP=AP+BP=2AP$

A circle touches the side $BC$ of $\triangle\ ABC$ at $P$ and touches $AB$ and $AC$ produced at $Q$ and $R$ respectively. Prove that $AQ=\dfrac{1}{2}$ (perimeter of $\triangle\ ABC$ ).

$DA$ and $DB$ are the triangle drwan from D, to the circle with centre $O$ , at A and B respectively $\angle ADO = 35^\circ$ find $\angle AOB$

Pair of tangents are drawn from point (1,3) to circle ${ x }^{ 2 }+{ y }^{ 2 }-3x-x+3y-1=0$ Find line joining their point of contact

Draw two concentric circles of radii $3 \,cm$ and $5 \,cm.$ Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre 0 and radius r at P. Q and R respectively.
Prove That :
(i) AB+CQ=AC+BQ
(ii) Area $(\Delta ABQ=\frac{1}{2}$ (Perimeter of $(\Delta ABC)\times r$

In the given figure, tangents $PQ$ and $PR$ are drawn from an external point p to a circle with center $O$ , such that $\angle RPQ=30^{\circ}$ . A chord $RS$ is drawn parallel to the tangent $PQ$ . Find $\angle RQS.$