## Selina Concise Mathematics Class 10 ICSE Solutions Tangents and Intersecting Chords

**Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 18 Tangents and Intersecting Chords**

### Tangents and Intersecting Chords Exercise 18A – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

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

**Solution:**

**Question 2.**

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.

**Solution:**

**Question 3.**

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.

**Solution:**

**Question 4.**

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.

**Solution:**

**Question 5.**

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.

**Solution:**

**Question 6.**

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.

**Solution:**

**Question 7.**

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

**Solution:**

**Question 8.**

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

**Solution:**

From A, AP and AS are tangents to the circle.

Therefore, AP = AS…….(i)

Similarly, we can prove that:

BP = BQ ………(ii)

CR = CQ ………(iii)

DR = DS ………(iv)

Adding,

AP + BP + CR + DR = AS + DS + BQ + CQ

AB + CD = AD + BC

Hence, AB + CD = AD + BC

But AB = CD and BC = AD…….(v) Opposite sides of a ||gm

Therefore, AB + AB = BC + BC

2AB = 2 BC

AB = BC ……..(vi)

From (v) and (vi)

AB = BC = CD = DA

Hence, ABCD is a rhombus.

**Question 9.
**From the given figure prove that:

AP + BQ + CR = BP + CQ + AR.

Also, show that AP + BQ + CR = \(\frac{1}{2}\) × perimeter of triangle ABC.

**Solution:**

**Question 10.
**In the figure, if AB = AC then prove that BQ = CQ.

**Solution:**

**Question 11.**

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.

**Solution:**

**Question 12.
**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 perpendicular bisector of chord AB.

**Solution:**

**Question 13.
**In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that:

i) tangent at point P bisects AB.

ii) Angle APB = 90°

**Solution:**

**Question 14.
**Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. Prove that:

∠PAQ = 2∠OPQ

**Solution:**

**Question 15.**

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

**Solution:**

**Question 16.**

ABC is a right angled 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.

**Solution:**

**Question 17.
**In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate:

i) ∠QOR

ii) ∠QPR

given that ∠A = 60°

**Solution:**

**Question 18.
**In the following figure, PQ and PR are tangents to the circle, with centre O. If , calculate:

i) ∠QOR

ii) ∠OQR

iii) ∠QSR

**Solution:**

**Question 19.**

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

**Solution:**

**Question 20.**

In quadrilateral ABCD, angle 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.

**Solution:**

**Question 21.
**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 and ∠SPR = x° and ∠QRP = y°

Prove that -;

i) ∠ORS = y°

ii) write an expression connecting x and y

**Solution:**

**Question 22.
**PT is a tangent to the circle at T. If ; calculate:

i) ∠CBT

ii) ∠BAT

iii) ∠APT

**Solution:**

**Question 23.**

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.

**Solution:**

**Question 24.
**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.

**Solution:**

∠CAB = ∠BAQ = 30°……(AB is angle bisector of ∠CAQ)

∠CAQ = 2∠BAQ = 60°……(AB is angle bisector of ∠CAQ)

∠CAQ + ∠PAC = 180°……(angles in linear pair)

∴∠PAC = 120°

∠PAC = 2∠CAD……(AD is angle bisector of ∠PAC)

∠CAD = 60°

Now,

∠CAD + ∠CAB = 60 + 30 = 90°

∠DAB = 90°

Thus, BD subtends 90° on the circle

So, BD is the diameter of circle

### Tangents and Intersecting Chords Exercise 18B – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

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.

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

**Solution:**

**Question 2.**

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.

**Solution:**

**Question 3.**

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:

i) ∠QAD

ii) ∠PAD

iii) ∠CDB

**Solution:**

**Question 4.**

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

i) ∠PRS

ii) ∠ROT

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

**Solution:**

**Question 5.**

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.

**Solution:**

**Question 6.**

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.

**Solution:**

**Question 7.**

Two circles with centers 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.

**Solution:**

**Question 8.**

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

**Solution:**

**Question 9.**

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.

**Solution:**

**Question 10.**

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:

i) angle BCT

ii) angle DOC

**Solution:**

**Question 11.**

Two circles intersect each other at point A and B. A straight line PAQ cuts the circle 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.

**Solution:**

**Question 12.**

In the figure, PA is a tangent to the circle. PBC is a secant and AD bisects angle BAC.

Show that the triangle PAD is an isosceles triangle. Also show that:

∠CAD = \(\frac{1}{2}\)(∠PBA – ∠PAB)

**Solution:**

**Question 13.**

Two circles intersect each other at point 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.

**Solution:**

**Question 14.**

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

i) if , ∠CDE = 90° AB = 5 cm, BD = 4 cm and CD = 9 cm; find DE

ii) If AD = BD, Show that AE = BC.

**Solution:**

**Question 15.**

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

**Solution:**

**Question 16.**

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

**Solution:**

### Tangents and Intersecting Chords Exercise 18C – Selina Concise Mathematics Class 10 ICSE Solutions

**Question 1.**

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

**Solution:**

**Question 2.
**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 \(32 \sqrt{3}\) cm

^{2}, find the radius of the circle.

**Solution:**

**Question 3.**

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.

**Solution:**

**Question 4.**

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

**Solution:**

**Question 5.**

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 centre of the circle?

**Solution:**

**Question 6.**

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.

**Solution:**

**Question 7.
**In the given figure, C and D are points on the semicircle described on AB as diameter.

Given angle BAD = 70° and angle DBC = 30°, calculate angle BDC

**Solution:**

**Question 8.**

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

**Solution:**

ABCD is a cyclic quadrilateral.

∴ ∠A + ∠C = 180°

⇒ 3∠C + ∠C = 180°

⇒ 4∠C = 180°

⇒ ∠C = 45°

∵ ∠A = 3∠C

⇒ ∠A = 3 × 45°

⇒ ∠A = 135°

Similarly,

∴ ∠B+ ∠D = 180°

⇒∠B + 5∠B = 180°

⇒ 6∠B = 180°

⇒ ∠B = 30°

∵∠D = 5∠B

⇒ ∠D = 5 × 30° >

⇒ ∠D = 150°

Hence, ∠A = 1350, ∠B = 30°, ∠C = 450, ∠D = 150°

**Question 9.**

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

**Solution:**

**Question 10.**

Bisectors of vertex A, B and C of a triangle ABC intersect its circumcircle at points D, E and F respectively. Prove that angle EDF = \(90^{\circ}-\frac{1}{2} \angle A\)

**Solution:**

**Question 11.**

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.

**Solution:**

**Question 12.**

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.

**Solution:**

**Question 13.**

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.

**Solution:**

**Question 14.
**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.

**Solution:**

**Question 15.
**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

**Solution:**

**Question 16.**

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

**Solution:**

**Question 17.
**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 × AD = AB

^{2 }ii) BD

^{2 }= AD × DC.

**Solution:**

**Question 18.
**In the given figure AC = AE.

Show that:

i) CP = EP

ii) BP = DP

**Solution:**

**Question 19.
**ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC=120°

Calculate:

i) ∠BEC

ii) ∠ BED

**Solution:**

**Question 20.
**In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30°, find:

(i) angle BCO

(ii) angle AOB

(iii) angle APB

**Solution:**

**Question 21.**

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 centres. Find the radii of the three circles.

**Solution:**

**Question 22.
**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 –

i) ∠ONL + ∠OML = 180°

ii) ∠BAM = ∠BMA

iii) ALOB is a cyclic quadrilateral.

**Solution:**

**Question 23.**

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

**Solution:**

**Question 24.
**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

ii) angle QRP

iii) angle QRS

iv) angle STR

**Solution:**

**Question 25.
**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.

**Solution:**

**Question 26.**

AB is a line segment and M is its midpoint. Three semicircles are drawn with AM, MB and AB as diameters on the same side of the line AB. A circle with radius r unit is drawn so that it touches all the three semicircles. Show that: AB = 6 x r

**Solution:**

**Question 27.**

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.

**Solution:**

**Question 28.**

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

**Solution:**

**Question 29.**

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

**Solution:**

**Question 30.**

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.

**Solution:**

**Question 31.**

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

**Solution:**

**Question 32.
**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.

**Solution:**

**Question 33.**

In the figure, XY is the diameter of the circle, PQ is the tangent to the circle at Y. Given that ∠AXB = 50° and ∠ABX = 70°. Calculate ∠BAY and ∠APY.

**Solution:**

**Question 34.
**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

**Solution:**

**Question 35.
**In the given figure, AB is the diameter. The tangent at C meets AB produced at Q.

If

∠CAB = 34°, find

i) ∠CBA

ii) ∠CQB

**Solution:**

**Question 36.
**In the given figure, O is the centre of the circle. The tangets at B and D intersect each other at point P.

If AB is parallel to CD and ∠ABC = 55°, find:

i) ∠BOD

ii) ∠BPD

**Solution:**

**Question 37.
**In the figure given below PQ =QR, ∠RQP = 68°, PC and CQ are tangents to the circle with centre O. Calculate the values of:

i) ∠QOP

ii) ∠QCP

**Solution:**

**Question 38.**

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

**Solution:**

**Question 39.
**In the figure, given below, AC is a transverse common tangent to two circles with centers P and Q and of radii 6 cm and 3 cm respectively.

Given that AB = 8 cm, calculate PQ.

**Solution:**

**Question 40.**

In the figure given below, O is the centre of the circum circle of triangle XYZ. Tangents at X and Y intersect at point T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.

**Solution:**

**Question 41.**

In the given figure, AE and BC intersect each other at point D. If ∠CDE=90°, AB = 5 cm, BD = 4 cm and CD = 9 cm, find AE.

**Solution:**

**Question 42.**

In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.

**Solution:**

**Question 43.**

In the figure given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z.

**Solution:**

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