#### Chapters

Chapter 2: Banking (Recurring Deposit Account)

Chapter 3: Shares and Dividend

Chapter 4: Linear Inequations (In one variable)

Chapter 5: Quadratic Equations

Chapter 6: Solving (simple) Problems (Based on Quadratic Equations)

Chapter 7: Ratio and Proportion (Including Properties and Uses)

Chapter 8: Remainder and Factor Theorems

Chapter 9: Matrices

Chapter 10: Arithmetic Progression

Chapter 11: Geometric Progression

Chapter 12: Reflection

Chapter 13: Section and Mid-Point Formula

Chapter 14: Equation of a Line

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter 16: Loci (Locus and Its Constructions)

Chapter 17: Circles

Chapter 18: Tangents and Intersecting Chords

Chapter 19: Constructions (Circles)

Chapter 20: Cylinder, Cone and Sphere

Chapter 21: Trigonometrical Identities

Chapter 22: Height and Distances

Chapter 23: Graphical Representation

Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)

Chapter 25: Probability

## Chapter 18: Tangents and Intersecting Chords

#### Exercise 18 (A) [Pages 274 - 276]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 18 Tangents and Intersecting Chords Exercise 18 (A) [Pages 274 - 276]

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

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

Two circle 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.

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

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

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

If the sides of a quadrilateral ABCD touch a circle, prove that:

AB + CD = BC + AD

If the sides of a parallelogram touch a circle in following figure , Prove that the parallelogram is a rhombus.

From the given figure, prove that:

AP + BQ + CR = BP + CQ + AR

Also show that:

AP + BQ + CR = `1/2`× Perimeter of ΔABC.

In the following figure; If AB = AC then prove that BQ = CQ.

Radii of two circles are 6. 3 cm and 3.6 cm. State the distance between their centres if:

they touch each other externally

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if -

they touch each other internally.

From a point P outside a circle, with centre O, tangents PA and PB are drawn. Prove that:

∠AOP = ∠BOP

From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:

OP is the perpendicular bisector of chord AB.

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that

tangent at point P bisects AB,

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that:

(ii) angles APB = 90°

Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. prove that: PAQ = 2 ∠OPQ

Two parallel tangents of a circle meet a third tangent at points P and Q. prove that PQ subtends a right angle at the centre.

ABC is a right angles triangle with AB = 12 cm and AC = 13 cm. A circle, with centre O, has been inscribed inside the triangle.

Calculate the value of x, the radius of the inscribed circle

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate : (i) `∠`QOR

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate:

i)`∠`QPR .

In the following figure, PQ and PR are tangents to the circle, with centre O. If `∠`QPR = 60°, calculate:

∠QOR

In the following figure, PQ and PR are tangents to the circle, with centre O. If ∠ QPR = 60° , calculate:

∠ OQR

In the following figure, PQ and PR are tangents to the circle, with centre O. If ∠ QPR = 60° , calculate:

∠ QSR

In the given figure, AB is the diameter of the circle, with centre O, and AT is the tangent. Calculate the numerical value of x.

In quadrilateral ABCD; angles D = 90°, BC = 38 cm and DC = 25 cm. A circle is inscribed in this quadrilateral which touches AB at point Q such that QB = 27 cm, Find the radius of the circle.

In the given figure, PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent TR at P.

Given ∠SPR = x° and ∠QRP = y°;

Prove that:

(i) ∠ORS = y°

(ii) Write an expression connecting x and y.

PT is a tangent to the circle at T. if ∠ABC = 70° and ∠ACB = 50°; Calculate:

∠ CBT

PT is a tangent to the circle at T. If ∠ ABC = 70° and ∠ ACB = 50° ; calculate : ∠ APT

PT is a tangent to the circle at T. If ∠ ABC = 70° and ∠ ACB = 50° ; calculate : ∠ APT

In the given figure, O is the centre of the circumcircle ABC. Tangents at A and C intersect at P. Given angle AOB = 140° and angle APC = 80°; find the angle BAC.

In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of ∠CAQ and ∠PAC. If ∠BAQ = 30°, prove that : BD is diameter of the circle.

#### Exercise 18 (B) [Pages 283 - 284]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 18 Tangents and Intersecting Chords Exercise 18 (B) [Pages 283 - 284]

i) In the given figure 3 × CP = PD = 9cm and AP = 4.5 cm Find BP.

(ii) In the given figure, 5 × PA = 3 × AB = 30 cm and PC = 4cm. find CD.

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

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 AB.

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 the length of tangent PT.

In the following figure, PQ is the tangent to the circle at A, DB is the diameter and O is the centre of the circle. If ∠ADB = 30° and ∠CBD = 60°, Calculate: ∠QAB

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 = 30° and ∠ CBD = 60° ; calculate ∠ PAD.

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 = 30° and ∠ CBD = 60° ; calculate : ∠CDB

If PQ is a tangent to the circle at R; calculate:

(i) ∠PRS

(ii) ∠ROT

Given : O is the centre of the circle and angle TRQ = 30°

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

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

Two circle 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.

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.

In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle BCD. Prove that the diagonal BD is parallel to the tangent to the circle at point A.

In the figure, ABCD is a cyclic quadrilateral with BC = CD. TC is tangent to the circle at point C and DC is produced to point G. If ∠BCG = 108° and O is the centre of the circle, find : Angle BCT

In the figure, ABCD is a cyclic quadrilateral with BC = CD. TC is tangent to the circle at point C and DC is produced to point G. If angle BCG=108° and O is the centre of the circle, find: angle DOC

Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; show that the points P, B, Q and T are concyclic

In the figure; PA is a tangent to the circle, PBC is secant and AD bisects angle BAC. Show that triangle PAD is an isosceles triangle. Also, show that:

`∠CAD =1/2(∠PBA-∠PAB)`

Two circles intersect each other at points A and B. their common tangent touches the circles at points P and Q as shown in the figure. Show that the angles PAQ and PBQ are supplementary.

In the figure, chords AE and BC intersect each other at point D.

If ∠ CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm

Find DE.

In the figure, chords AE and BC intersect each other at point D.

If AD = BD, Show that AE = BC.

Circles with centres P and Q intersect at points A and B as shown in the figure. CBD is a segment and EBM is tangent to the circle with centre Q, at point B. If the circle are congruent; show that

CE = BD

In the adjoining figure, O is the centre of the circle and AB is a tangent to it at point B. ∠BDC = 65° Find ∠BAO.

#### Exercise 18 (C) [Pages 285 - 288]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 18 Tangents and Intersecting Chords Exercise 18 (C) [Pages 285 - 288]

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

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

OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O.

(ii) If the area of the rhombus is 32 3 cm2 find the radius of the circle.

Two circle with centres 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.

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

The diameter and a chord of a 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 centre of the circle?

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

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

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

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

Bisectors of vertex angles A, B, and C of a triangle ABC intersect its circumcircle at the points D, E and F respectively. Prove that angle EDF = 90° - `1/2` `∠`A

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 = 20°, find angle AOD.

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.

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

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.

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 = 40° and ∠ABD = 60°, find;

(i) ∠DBC (ii) ∠BCP (iii) ∠ADB

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

ABC is a right triangle with angle B = 90°, A circle with BC as diameter meets hypotenuse AC at point D. prove that:

AC × AD = `"AB"^2`

ABC is a right triangle with angle B = 90º. A circle with BC as diameter meets by hypotenuse AC at point D.

Prove that - BD^{2} = AD x DC.

In the given figure, AC = AE Show that:

(i) CP = EP

(ii) BP = DP

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = 120°

Calculate : ∠BEC

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC=120°.

Calculate: ∠ BED.

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

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

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

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

In a square ABCD, its diagonals AC and BD intersect each other at point O. the bisector of angle DAO meets BD at point M and the bisector of angle ABD meets AC at N and AM at L. Show that:

∠ ONL + ∠ OML = 180

In a square ABCD, its diagonal AC and BD intersect each other at point O. The bisector of angle DAO meets BD at point M and bisector of angle ABD meets AC at N and AM at L. Show that - ∠ BAM = ∠ BMA

In a square ABCD, its diagonal AC and BD intersect each other at point O. The bisector of angle DAO meets BD at point M and bisector of angle ABD meets AC at N and AM at L. Show that - ALOB is a cyclic quadrilateral.

The given figure shows a semi-circle with centre O and diameter PQ. If PA = AB and ∠BCQ =140°; Find measures of angles PAB and AQB. Also, show that AO is parallel to BQ.

The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:

(i) angle QTR

The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:

(ii) angle QRP

The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:

(iii) angle QRS

(iv) angle STR

In the given figure, PAT is tangent to the circle with centre O at point A on its circumference and is parallel to chord BC. If CDQ is a line segment , show that:

(i) ∠BAP = ∠ADQ

(ii) ∠AOB = 2∠ADQ

(iii) ∠ADQ = ∠ADB

TA and TB are tangents to a circle with centre O from an external point T. OT intersects the circle at point P. Prove that AP bisects the angle TAB.

Two circles intersect in points P and Q. A secant passing through P intersects the circles in Aand B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, B and T lie on a circle.

Prove that any four vertices of a regular pentagon are concylic (lie on the same circle)

Chords AB and CD of a circle when extended meet at point X. Given AB = 4 cm, BX = 6 cm and XD = 5 cm, calculate the length of CD.

In the given figure, find TP if AT = 16 cm and AB = 12 cm.

In the following figure, a circle is inscribed in the quadrilateral ABCD.

If BC = 38 cm, QB = 27 cm, DC = 25 cm and that AD is perpendicular to DC, find the radius of the circle.

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.

If ∠AXB = 50° and ∠ABX = 70° and ∠BAY and ∠APY

In the given figure, QAP is the tangent at point A and PBD is a straight line.

If ∠ACB = 36° and ∠APB = 42°, find:

(i) ∠BAP (ii) ∠ABD (iii) ∠QAD (iv) ∠BCD

In the given figure, AB is the diameter. The tangent at C meets AB produced at Q.

If ∠CAB = 34°, Find : ∠CBA

In the given figure, AB is the diameter. The tangent at C meets AB produced at Q.

If ∠ CAB = 34° , find : ∠ CQB

In the given figure, O is the centre of the circle. The tangents at B and D intersect each other at point P. If AB is parallel to CD and ∠ABC = 55°, find:

(i) ∠BOD (ii) ∠BPD

In two concentric circles, prove that all chords of the outer circle, which touch the inner circle, are of equal length.

In the following figure, PQ = QR, `∠`RQP = 68° , PC and CQ are tangents to the circle with centre O

(i) `∠`QOP

(ii) `∠`QCP

In the figure, given below, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively.

Given that AB = 8 cm, calculate PQ.

In the figure, given below, O is the centre of the circumcircle of triangle XYZ.

Tangents at X and Y intersect at point T. Given ∠XTY = 80°, and ∠XOZ = 140°, calculate the value of ∠ZXY.

In the given figure, AE and BC intersect each other as point D.

If ∠CDE = 90°, AB = 5cm, BD = 4cm and CD = 9 cm find AE.

In the given circle with center o, ∠ABC=100°, ∠ACD=40° and CT is tangent to the circle at C. find ∠ADC and ∠DCT.

In the figure given below, O is the center of the circle of the circle and SP is a tangent. if ∠SRT=65°, find the value of x, y and Z.

## Chapter 18: Tangents and Intersecting Chords

## Selina solutions for Concise Mathematics Class 10 ICSE chapter 18 - Tangents and Intersecting Chords

Selina solutions for Concise Mathematics Class 10 ICSE chapter 18 (Tangents and Intersecting Chords) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Concise Mathematics Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Selina textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Concise Mathematics Class 10 ICSE chapter 18 Tangents and Intersecting Chords are Tangent to a Circle, Number of Tangents from a Point on a Circle, Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord, Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof), Chord Properties - Equal Chords Are Equidistant from the Center, Chord Properties - Chords Equidistant from the Center Are Equal (Without Proof), Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line, Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle, Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof), Arc and Chord Properties - Angle in a Semi-circle is a Right Angle, Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse, Arc and Chord Properties - If Two Chords Are Equal, They Cut off Equal Arcs, and Its Converse (Without Proof), Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal, Cyclic Properties, Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior., Areas of Sector and Segment of a Circle, Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments, Tangent Properties - If a Chord and a Tangent Intersect Externally, Then the Product of the Lengths of Segments of the Chord is Equal to the Square of the Length of the Tangent from the Point of Contact to the Point of Intersection, Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.

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