Tangents and Intersecting Chords Archives

ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centers. Find the radii of the three circles.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: According to the given question, ABC is a triangle with \[AB\text{ }=\text{ }10\text{ }cm,\text{ }BC=\text{ }8\text{ }cm,\text{ }AC\text{ }=\text{ }6\text{ }cm\] Three circles are drawn...

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30degree, find: angle APB

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: In the given fig, \[O\]is the centre of the circle and, \[CA\text{ }and\text{ }CB\]are the tangents to the circle from \[C\]Also, \[\angle ACO\text{ }=\text{ }{{30}^{o}}\] \[P\] is any...

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30degree, find: (i) angle BCO (ii) angle AOB

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: In the given fig, \[O\]is the centre of the circle and, \[CA\text{ }and\text{ }CB\]are the tangents to the circle from \[C\]Also, \[\angle ACO\text{ }=\text{ }{{30}^{o}}\] \[P\] is any...

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = 120degree Calculate: i) ∠BEC ii) ∠BED

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

i) Join \[OC\text{ }and\text{ }OB\] \[AB\text{ }=\text{ }BC\text{ }=\text{ }CD\] And \[\angle ABC\text{ }=\text{ }{{120}^{o}}~\][Given] So, \[\angle BCD\text{ }=\angle ABC\text{ }=\text{...

In the given figure, AC = AE. Show that: i) CP = EP ii) BP = DP

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: In \[\vartriangle ADC\text{ }and\text{ }\vartriangle ABE\] \[\angle ACD\text{ }=\angle AEB~\] [Angles in the same segment] \[AC\text{ }=\text{ }AE\][Given] \[\angle A\text{ }=\angle...

ABC is a right triangle with angle B = 90º. A circle with BC as diameter meets by hypotenuse AC at point D. Prove that: i) AC x AD = AB^2 ii) BD^2 = AD x DC.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

i) In \[\vartriangle ABC\] We know that \[\angle B\text{ }=\text{ }{{90}^{o}}~and\text{ }BC\]is the diameter of the circle. Hence, \[AB\]is the tangent to the circle at \[B\] Now, as \[AB\] is...

The given figure shows a circle with centre O and BCD is a tangent to it at C. Show that: ∠ACD + ∠BAC = 90degree

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Join \[OC\] \[BCD\]is the tangent and \[OC\]is the radius. As, \[OC\bot BD\] \[\angle OCD\text{ }=\text{ }{{90}^{o}}\] \[\angle OCD\text{ }+\angle ACD\text{ }=\text{ }{{90}^{o}}~\ldots...

In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40degree and ∠ABD = 60degree, find: ∠ADB

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: In \[\vartriangle ABD\] \[\angle BAD\text{ }=\text{ }{{90}^{o}}\] [Angle in a semi-circle], \[\angle ABD\text{ }=\text{ }{{60}^{o}}~\] [Given] Or, \[\angle ADB\text{ }=\text{...

In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40degree and ∠ABD = 60degree, find: i) ∠DBC ii) ∠BCP

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: 15. (i) \[PQ\]is a tangent and \[CD\]is a chord. \[\angle DCQ\text{ }=\angle DBC\] [Angles in the alternate segment] \[\angle DBC\text{ }=\text{ }{{40}^{o}}\] \[[As\angle DCQ\text{...

In the given figure, MN is the common chord of two intersecting circles and AB is their common tangent. Prove that the line NM produced bisects AB at P.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: From \[P,\text{ }AP\] is the tangent And \[PMN\]is the secant for first circle. \[A{{P}^{2}}~=\text{ }PM\text{ }x\text{ }PN\text{ }\ldots .\text{ }\left( 1 \right)\] Again from \[P,\text{...

P is the midpoint of an arc APB of a circle. Prove that the tangent drawn at P will be parallel to the chord AB.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[AP\text{ }and\text{ }BP\] As \[TPS\] is a tangent And \[PA\]is the chord of the circle. \[\angle BPT\text{ }=\angle PAB~\] [Angles in alternate segments] But, \[\angle PBA\text{ }=\angle...

Prove that the perimeter of a right triangle is equal to the sum of the diameter of its incircle and twice the diameter of its circumcircle.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[OL,\text{ }OM\text{ }and\text{ }ON\] Suppose, \[D\text{ }and\text{ }d\] be the diameter of the circumcircle and incircle. Also, \[R\text{ }and\text{ }r\] be the radius of the circumcircle and...

In the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. If ∠C = 20degree, find angle AOD.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Join \[OB\] In \[\vartriangle OBC\] we have \[BC\text{ }=\text{ }OD\text{ }=\text{ }OB\][Radii of the same circle] \[\angle BOC\text{ }=\angle BCO\text{ }=\text{ }{{20}^{o}}\] And ext....

Bisectors of vertex angles A, B and C of a triangle ABC intersect its circumcircle at points D, E and F respectively. Prove that angle EDF = 90degree – ½ ∠A

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[ED,\text{ }EF\text{ }and\text{ }DF\] And \[BF,\text{ }FA,\text{ }AE\text{ }and\text{ }EC\] \[\angle EBF\text{ }=\angle ECF\text{ }=\angle EDF\text{ }\ldots ..\text{ }\left( i \right)\] [Angle...

Show that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[AD\] And \[AB\] is the diameter. We have \[\angle ADB\text{ }=\text{ }90{}^\text{o}~\][Angle in a semi-circle] But, \[\angle ADB\text{ }+~\angle ADC\text{ }=\text{ }180{}^\text{o}~\][Linear...

In cyclic quadrilateral ABCD, ∠A = 3 ∠C and ∠D = 5 ∠B. Find the measure of each angle of the quadrilateral.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Given, cyclic quadrilateral \[ABCD\] So, \[\angle A\text{ }+\angle C\text{ }=\text{ }{{180}^{o}}~\][Opposite angles in a cyclic quadrilateral is supplementary] \[3\angle C\text{ }+\angle C\text{...

In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = 70degree and angle DBC = 30degree, calculate angle BDC.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: \[ABCD\] is a cyclic quadrilateral, \[\angle BCD\text{ }+~\angle BAD\text{ }=\text{ }{{180}^{\circ }}\] [Opposite angles of a cyclic quadrilateral are supplementary] \[\angle BCD\text{...

ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110o and angle BAC = 50o. Find angle DAC and angle DCA.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Given, \[ABCD\] is a cyclic quadrilateral in which \[AD\text{ }||\text{ }BC\] And, \[\angle ADC\text{ }=\text{ }{{110}^{o}},\angle BAC\text{ }=\text{ }{{50}^{o}}\] We know that, \[\angle B\text{...

The diameter and a chord of circle have a common end-point. If the length of the diameter is 20 cm and the length of the chord is 12 cm, how far is the chord from the center of the circle?

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

We have, \[AB\]as the diameter and \[AC\]as the chord. Now, draw \[OL\bot AC\] Since\[OL\bot AC\] and hence it bisects \[AC\] \[O\] is the centre of the circle. Therefore, \[OA\text{ }=\text{...

Two chords AB and AC of a circle are equal. Prove that the center of the circle, lies on the bisector of the angle BAC.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Given: \[AB\text{ }and\text{ }AC\] are two equal chords of \[C\text{ }\left( O,\text{ }r \right)\] Required to prove: Centre, \[O\] lies on the bisector of \[\angle BAC\] Construction: Join \[BC\]...

Two circles with centers A and B, and radii 5 cm and 3 cm, touch each other internally. If the perpendicular bisector of the segment AB meets the bigger circle in P and Q; find the length of PQ.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

We know that, If two circles touch internally, then distance between their centres is equal to the difference of their radii. So, \[AB\text{ }=\text{ }\left( 5\text{ }-\text{ }3 \right)\text{...

OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O. i) If the radius of the circle is 10 cm, find the area of the rhombus. ii) If the area of the rhombus is 32√3 cm2, find the radius of the circle.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

(i) Given, \[radius\text{ }=\text{ }10\text{ }cm\] In rhombus \[OABC\] \[OC\text{ }=\text{ }10\text{ }cm\] So,...

Prove that, of any two chords of a circle, the greater chord is nearer to the center.

Class 10, Exercise 18C, ICSE, Maths, Selina, Tangents and Intersecting Chords

Given: \[A\] circle with center \[O\] and radius \[r\] \[AB\text{ }and\text{ }CD\]are two chords such that \[AB\text{ }>\text{ }CD\] Also, \[OM\bot AB\text{ }and\text{ }ON\bot CD\] Required to...

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Prove that: ∠CPA = ∠DPB

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Let’s draw a tangent \[TS\text{ }at\text{ }P\]to the circles given. As \[TPS\]is the tangent and \[PD\]is the chord, we have \[\angle PAB\text{ }=\angle BPS\text{ }\ldots .\text{ }\left( i...

Two circles with centres O and O’ are drawn to intersect each other at points A and B. Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O’ at A. Prove that OA bisects angle BAC.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Join \[OA,\text{ }OB,\text{ }OA,\text{ }OB\text{ }and\text{ }OO\] \[CD\]is the tangent and \[AO\]is the chord. \[\angle OAC\text{ }=\angle OBA\text{ }\ldots \text{ }\left( i \right)\] ...

Tangent at P to the circumcircle of triangle PQR is drawn. If this tangent is parallel to side QR, show that triangle PQR is isosceles.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Let \[DE\]be the tangent to the circle at \[P\] And, \[DE\text{ }||\text{ }QR\][Given] \[\angle EPR\text{ }=\angle PRQ\][Alternate angles are equal] \[\angle DPQ\text{ }=\angle PQR\][Alternate...

AB is diameter and AC is a chord of a circle with centre O such that angle BAC=30º. The tangent to the circle at C intersects AB produced in D. Show that BC = BD.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[OC\] \[\angle BCD\text{ }=\angle BAC\text{ }=\] \[{{30}^{o}}~\] [Angles in the alternate segment] It’s seen that, arc \[BC\]subtends \[\angle DOC\]at the center of the circle And \[\angle...

If PQ is a tangent to the circle at R; calculate: i) ∠PRS ii) ∠ROT When O is the centre of the circle and ∠TRQ = 30degree

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) As \[PQ\] is the tangent and \[OR\]is the radius. So, \[OR\bot PQ\] \[\angle ORT\text{ }=\text{ }{{90}^{o}}\] \[\angle TRQ\text{ }=\text{ }{{90}^{o}}-\text{ }{{30}^{o}}~=\text{...

In the following figure, PQ is the tangent to the circle at A, DB is a diameter and O is the centre of the circle. If ∠ADB = 30degree and ∠CBD = 60degree; calculate: ∠CDB

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: As \[BD\] is the diameter, we have \[\angle BCD\text{ }=\text{ }{{90}^{o}}\] [Angle in a semi-circle] Now in \[\vartriangle BCD\] \[\angle CDB\text{ }+\angle CBD\text{ }+\angle BCD\text{...

In the following figure, PQ is the tangent to the circle at A, DB is a diameter and O is the centre of the circle. If ∠ADB = 30degree and ∠CBD = 60degree; calculate: i) ∠QAB ii) ∠PAD

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) Given, \[PAQ\]is a tangent and \[AB\]is the chord \[\angle QAB~=\angle ADB\text{ }=\text{ }{{30}^{o}}~\] [Angles in the alternate segment] (ii) \[OA\text{ }=\text{ }OD\][radii of the...

In the given figure, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find (i) AB. (ii) the length of tangent PT.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) \[PA\text{ }=\text{ }AB\text{ }+\text{ }BP\text{ }=\text{ }\left( AB\text{ }+\text{ }4 \right)\text{ }cm\] \[PC\text{ }=\text{ }PD\text{ }+\text{ }CD\] \[=\text{ }5\text{ }+\text{...

In the given figure, tangent PT = 12.5 cm and PA = 10 cm; find AB.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: As \[PAB\]is the secant and \[PT\]is the tangent, we have \[P{{T}^{2}}~=\text{ }PA\text{ }x\text{ }PB\] \[{{12.5}^{2}}~=\text{ }10\text{ }x\text{ }PB\] \[PB\text{ }=\text{ }\left(...

(i) In the given figure, 3 x CP = PD = 9 cm and AP = 4.5 cm. Find BP. (ii) In the given figure, 5 x PA = 3 x AB = 30 cm and PC = 4cm. Find CD.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

(i) (ii) Solution: (i) As the two chords \[AB\text{ }and\text{ }CD\] intersect each other at \[P\] We have \[AP\text{ }x\text{ }PB\text{ }=\text{ }CP\text{ }x\text{ }PD\] \[4.5\text{ }x\text{...

From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that: i) ∠AOP = ∠BOP ii) OP is the ⊥ bisector of chord AB.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

According to the given question, i) In \[\vartriangle AOP\text{ }and\text{ }\vartriangle BOP\], we have \[AP\text{ }=\text{ }BP\] \[\left[ Tangents\text{ }from\text{ }P\text{ }to\text{ }the\text{...

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if – i) they touch each other externally. ii) they touch each other internally.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

According to the given question \[Radius\text{ }of\text{ }bigger\text{ }circle\text{ }=\text{ }6.3\text{ }cm\]and \[smaller\text{ }circle\text{ }radius\text{ }=\text{ }3.6\text{ }cm\] i) The two...

In the figure, if AB = AC then prove that BQ = CQ.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: From point \[A\], we know that \[AP\text{ }and\text{ }AR\]are the tangents to the circle So, we have \[AP\text{ }=\text{ }AR\] Similarly, we also have \[BP\text{ }=\text{ }BQ\text{...

From the given figure prove that: AP + BQ + CR = BP + CQ + AR. Also, show that AP + BQ + CR = ½ x perimeter of triangle ABC.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: From point \[B,\text{ }BQ\text{ }and\text{ }BP\]are the tangents to the circle We have, \[BQ\text{ }=\text{ }BP\text{ }\ldots \ldots \ldots ..\left( 1 \right)\] Similarly, we also get...

If the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Suppose a circle touch the sides \[AB,\text{ }BC,\text{ }CD\text{ }and\text{ }DA\]of parallelogram \[ABCD\text{ }at\text{ }P,\text{ }Q,\text{ }R\text{ }and\text{ }S\]respectively. Now,...

If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Suppose, a circle touch the sides \[AB,\text{ }BC,\text{ }CD\text{ }and\text{ }DA\] of quadrilateral \[ABCD\text{ }at\text{ }P,\text{ }Q,\text{ }R\text{ }and\text{ }S\]respectively. As,...

Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Suppose, \[ABC\]be the triangle formed when centres of 3 circles are joined. Given, \[AB\text{ }=\text{ }6\text{ }cm,\text{ }AC\text{ }=\text{ }8\text{ }cm\text{ }and\text{ }BC\text{ }=\text{...

Two circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Given, \[OS\text{ }=\text{ }5\text{ }cm\text{ }and\text{ }OT\text{ }=\text{ }3\text{ }cm\] In right triangle\[OST\], we have \[S{{T}^{2}}~=\text{ }O{{S}^{2}}~\text{ }O{{T}^{2}}\] \[=\text{...

Two circles touch each other internally. Show that the tangents drawn to the two circles from any point on the common tangent are equal in length.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Suppose, \[Q\]be the point on the common tangent from which, two tangents \[QA\text{ }and\text{ }QP\]are drawn to the circle with \[centre\text{ }O.\] So, \[QA\text{ }=\text{ }QP\text{...

Two circles touch each other externally at point P. Q is a point on the common tangent through P. Prove that the tangents QA and QB are equal.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Suppose, \[Q\]be the point from which, \[QA\text{ }and\text{ }QP\]are two tangents with centre \[O\] So, \[QA\text{ }=\text{ }QP\text{ }\ldots ..\left( a \right)\] Similarly, from point...

In the given figure, O is the centre of the circle and AB is a tangent to the circle at B. If AB = 15 cm and AC = 7.5 cm, calculate the radius of the circle.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: Given, \[AB\text{ }=\text{ }15\text{ }cm,\text{ }AC\text{ }=\text{ }7.5\text{ }cm\] Suppose,the radius of the circle to be \[r.\] So, \[AO\text{ }=\text{ }AC\text{ }+\text{ }OC\text{...

The radius of a circle is 8cm. Calculate the length of a tangent drawn to this circle from a point at a distance of 10cm from its centre.

Class 10, Exercise 18A, ICSE, Maths, Selina, Tangents and Intersecting Chords

Given, a circle with \[centre\text{ }O\]and \[radius\text{ }8\text{ }cm.\] An external point \[P\]is the point, from where a tangent is drawn to meet the circle at \[T.\] \[OP\text{ }=\text{...