{"id":15797,"date":"2024-02-29T05:06:54","date_gmt":"2024-02-28T23:36:54","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=15797"},"modified":"2024-02-29T15:00:09","modified_gmt":"2024-02-29T09:30:09","slug":"selina-icse-solutions-class-10-maths-tangents-intersecting-chords","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/selina-icse-solutions-class-10-maths-tangents-intersecting-chords\/","title":{"rendered":"Selina Concise Mathematics Class 10 ICSE Solutions Tangents and Intersecting Chords"},"content":{"rendered":"

Selina Concise Mathematics Class 10 ICSE Solutions Tangents and Intersecting Chords<\/h2>\n

Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 18 Tangents and Intersecting Chords<\/strong><\/p>\n

Tangents and Intersecting Chords Exercise 18A –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n

Question 1.<\/strong><\/span>
\nThe 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?
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 2.<\/strong><\/span><\/p>\n

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.
\n\"Selina
\nSolution:<\/strong><\/p>\n

\"Selina<\/p>\n

Question 3.<\/strong><\/span>
\nTwo 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.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 4.<\/strong><\/span>
\nTwo 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 5.<\/strong><\/span>
\nTwo circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 6.<\/strong><\/span>
\nThree 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 7.<\/strong><\/span>
\nIf the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 8.<\/strong><\/span>
\nIf the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

From A, AP and AS are tangents to the circle.
\nTherefore, AP = AS…….(i)
\nSimilarly, we can prove that:
\nBP = BQ ………(ii)
\nCR = CQ ………(iii)
\nDR = DS ………(iv)
\nAdding,
\nAP + BP + CR + DR = AS + DS + BQ + CQ
\nAB + CD = AD + BC
\nHence, AB + CD = AD + BC
\nBut AB = CD and BC = AD…….(v) Opposite sides of a ||gm
\nTherefore, AB + AB = BC + BC
\n2AB = 2 BC
\nAB = BC ……..(vi)
\nFrom (v) and (vi)
\nAB = BC = CD = DA
\nHence, ABCD is a rhombus.<\/p>\n

Question 9.
\n<\/strong><\/span>From the given figure prove that:
\nAP + BQ + CR = BP + CQ + AR.
\n\"Selina
\nAlso, show that AP + BQ + CR = \\(\\frac{1}{2}\\)\u00a0\u00a0\u00d7\u00a0perimeter of triangle ABC.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 10.
\n<\/strong><\/span>In the figure, if AB = AC then prove that BQ = CQ.
\n\"Selina
\nSolution:<\/strong><\/p>\n

\"Selina<\/p>\n

Question 11.<\/strong><\/span>
\nRadii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if –
\ni) they touch each other externally.
\nii) they touch each other internally.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 12.
\n<\/strong><\/span>From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:
\ni)\u00a0\u2220AOP =\u00a0\u2220BOP
\nii) OP is the perpendicular bisector of chord AB.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 13.
\n<\/strong><\/span>In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that:
\n\"Selina
\ni) tangent at point P bisects AB.
\nii) Angle APB = 90\u00b0
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 14.
\n<\/strong><\/span>Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. Prove that:
\n\u2220PAQ = 2\u2220OPQ
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 15.<\/strong><\/span>
\nTwo parallel tangents of a circle meet a third tangent at point P and Q. Prove that PQ subtends a right angle at the centre.
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 16.<\/strong><\/span>
\nABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with centre O, has been inscribed inside the triangle.
\n\"Selina
\nCalculate the value of x, the radius of the inscribed circle.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 17.
\n<\/strong><\/span>In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate:
\ni)\u00a0\u2220QOR
\nii)\u00a0\u2220QPR
\ngiven that\u00a0\u00a0\u2220A = 60\u00b0
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 18.
\n<\/strong><\/span>In the following figure, PQ and PR are tangents to the circle, with centre O. If\u00a0, calculate:
\ni)\u00a0\u2220QOR
\nii)\u00a0\u2220OQR
\niii)\u00a0\u2220QSR
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 19.<\/strong><\/span>
\nIn the given figure, AB is a diameter of the circle, with centre O, and AT is a tangent. Calculate the numerical value of x.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 20.<\/strong><\/span>
\nIn quadrilateral ABCD, angle D = 90\u00b0, 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 21.
\n<\/strong><\/span>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.
\nGiven\u00a0\u00a0and\u00a0\u2220SPR = x\u00b0 and\u00a0\u2220QRP = y\u00b0
\nProve that -;
\ni)\u00a0\u2220ORS = y\u00b0
\nii) write an expression connecting x and y
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 22.
\n<\/strong><\/span>PT is a tangent to the circle at T. If\u00a0; calculate:
\ni)\u00a0\u2220CBT
\nii)\u00a0\u2220BAT
\niii)\u00a0\u2220APT
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 23.<\/strong><\/span>
\nIn the given figure, O is the centre of the circumcircle ABC. Tangents at A and C intersect at P. Given angle AOB = 140\u00b0 and angle APC = 80\u00b0; find the angle BAC.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 24.
\n<\/strong><\/span>In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of\u00a0\u2220CAQ and\u00a0\u2220PAC. If\u00a0\u2220BAQ = 30\u00b0, prove\u00a0that :<\/span>\u00a0BD is diameter of the circle.
\n\"Selina
\nSolution:
\n<\/strong><\/span>\u2220CAB =\u00a0\u2220BAQ = 30\u00b0\u2026\u2026(AB is angle bisector of\u00a0\u2220CAQ)
\n\u2220CAQ = 2\u2220BAQ = 60\u00b0\u2026\u2026(AB is angle bisector of\u00a0\u2220CAQ)
\n\u2220CAQ +\u00a0\u2220PAC = 180\u00b0\u2026\u2026(angles in linear pair)
\n\u2234\u2220PAC = 120\u00b0
\n\u2220PAC = 2\u2220CAD\u2026\u2026(AD is angle bisector of\u00a0\u2220PAC)
\n\u2220CAD = 60\u00b0<\/p>\n

Now,
\n\u2220CAD +\u00a0\u2220CAB = 60 + 30 = 90\u00b0
\n\u2220DAB = 90\u00b0
\nThus, BD subtends 90\u00b0\u00a0on the circle
\nSo, BD is the diameter of circle<\/p>\n

Tangents and Intersecting Chords Exercise 18B –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n

Question 1.<\/strong><\/span>
\ni) In the given figure, 3 x CP = PD = 9 cm and AP = 4.5 cm. Find BP.
\n\"Selina
\nii) In the given figure, 5 x PA = 3 x AB = 30 cm and PC = 4cm. Find CD.
\n\"Selina
\niii) In the given figure, tangent PT = 12.5 cm and PA = 10 cm; find AB.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 2.<\/strong><\/span>
\nIn 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
\n(i) AB.
\n(ii) the length of tangent PT.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 3.<\/strong><\/span>
\nIn 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 ; \u2220ADB = 30\u00b0 and \u2220CBD = 60\u00b0 calculate:
\ni) \u2220QAD
\nii) \u2220PAD
\niii) \u2220CDB
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 4.<\/strong><\/span>
\nIf PQ is a tangent to the circle at R; calculate:
\ni) \u2220PRS
\nii) \u2220ROT
\n\"Selina
\nGiven: O is the centre of the circle and \u2220TRQ = 30\u00b0
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 5.<\/strong><\/span>
\nAB is diameter and AC is a chord of a circle with centre O such that angle BAC=30\u00ba. The tangent to the circle at C intersects AB produced in D. Show that BC = BD.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 6.<\/strong><\/span>
\nTangent at P to the circumcircle of triangle PQR is drawn. If this tangent is parallel to side QR, show that triangle PQR is isosceles.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 7.<\/strong><\/span>
\nTwo circles with centers O and O’ are drawn to intersect each other at points A and B.
\nCentre 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.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 8.<\/strong><\/span>
\nTwo 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: \u2220CPA = \u2220DPB
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 9.<\/strong><\/span>
\nIn 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 10.<\/strong><\/span>
\nIn 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\u00b0 and O is the centre of the circle, find:
\ni) angle BCT
\nii) angle DOC
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 11.<\/strong><\/span>
\nTwo 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 12.<\/strong><\/span>
\nIn the figure, PA is a tangent to the circle. PBC is a secant and AD bisects angle BAC.
\nShow that the triangle PAD is an isosceles triangle. Also show that:
\n\u2220CAD = \\(\\frac{1}{2}\\)(\u2220PBA – \u2220PAB)
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 13.<\/strong><\/span>
\nTwo 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.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 14.<\/strong><\/span>
\nIn the figure, chords AE and BC intersect each other at point D.
\ni) if , \u2220CDE = 90\u00b0 AB = 5 cm, BD = 4 cm and CD = 9 cm; find DE
\nii) If AD = BD, Show that AE = BC.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 15.<\/strong><\/span>
\nCircles 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.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 16.<\/strong><\/span>
\nIn the adjoining figure, O is the centre of the circle and AB is a tangent to it at point B. Find \u2220BDC = 65. Find \u2220BAO
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Tangents and Intersecting Chords Exercise 18C –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n

Question 1.<\/strong><\/span>
\nProve that of any two chord of a circle, the greater chord is nearer to the centre.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 2.
\n<\/strong><\/span>OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O.
\ni) If the radius of the circle is 10 cm, find the area of the rhombus.
\nii) If the area of the rhombus is \\(32 \\sqrt{3}\\) cm2<\/sup>, find the radius of the circle.
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 3.<\/strong><\/span>
\nTwo 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 4.<\/strong><\/span>
\nTwo chords AB and AC of a circle are equal. Prove that the centre of the circle, lies on the bisector of the angle BAC.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 5.<\/strong><\/span>
\nThe 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?
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 6.<\/strong><\/span>
\nABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110\u00b0 and angle BAC = 50\u00b0. Find angle DAC and angle DCA.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 7.
\n<\/strong><\/span>In the given figure, C and D are points on the semicircle described on AB as diameter.
\nGiven angle BAD = 70\u00b0 and angle DBC = 30\u00b0, calculate angle BDC
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 8.<\/strong><\/span>
\nIn cyclic quadrilateral ABCD,\u00a0\u2220<\/span>A = 3 \u2220C and\u00a0\u2220D = 5\u2220B. Find the measure of each angle of the quadrilateral.<\/span>
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

ABCD is a cyclic quadrilateral.
\n\u2234 \u2220A + \u2220C = 180\u00b0
\n\u21d2 3\u2220C + \u2220C = 180\u00b0
\n\u21d2 4\u2220C = 180\u00b0
\n\u21d2 \u2220C = 45\u00b0<\/p>\n

\u2235 \u2220A = 3\u2220C
\n\u21d2 \u2220A = 3 \u00d7 45\u00b0
\n\u21d2 \u2220A = 135\u00b0
\nSimilarly,<\/p>\n

\u2234 \u2220B+ \u2220D = 180\u00b0
\n\u21d2\u2220B + 5\u2220B = 180\u00b0
\n\u21d2 6\u2220B = 180\u00b0
\n\u21d2 \u2220B = 30\u00b0<\/p>\n

\u2235\u2220D = 5\u2220B
\n\u21d2 \u2220D = 5 \u00d7 30\u00b0 >
\n\u21d2 \u2220D = 150\u00b0
\nHence, \u2220A = 1350, \u2220B = 30\u00b0, \u2220C = 450, \u2220D = 150\u00b0<\/p>\n

Question 9.<\/strong><\/span>
\nShow that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 10.<\/strong><\/span>
\nBisectors 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\\)
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 11.<\/strong><\/span>
\nIn the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. If\u00a0\u2220C = 20\u00b0, find angle AOD.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 12.<\/strong><\/span>
\nProve 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 13.<\/strong><\/span>
\nP is the midpoint of an arc APB of a circle. Prove that the tangent drawn at P will be parallel to the chord AB.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 14.
\n<\/strong><\/span>In the given figure, MN is the common chord of two intersecting circles and AB is their common tangent.
\n\"Selina
\nProve that the line NM produced bisects AB at P.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 15.
\n<\/strong><\/span>In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If\u00a0\u2220DCQ = 40\u00b0 and\u00a0\u2220ABD = 60\u00b0, find:
\ni)\u00a0\u2220DBC
\nii)\u00a0\u2220 BCP
\niii)\u00a0\u2220 ADB
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 16.<\/strong><\/span>
\nThe given figure shows a circle with centre O and BCD is a tangent to it at C. Show that:\u00a0\u2220ACD +\u00a0\u2220BAC = 90\u00b0
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 17.
\n<\/strong><\/span>ABC is a right triangle with angle B = 90\u00ba. A circle with BC as diameter meets by hypotenuse AC at point D.
\nProve that –
\ni) AC \u00d7 AD = AB2
\n<\/sup>ii) BD2\u00a0<\/sup>= AD \u00d7 DC.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 18.
\n<\/strong><\/span>In the given figure AC = AE.
\nShow that:
\ni) CP = EP
\nii) BP = DP
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 19.
\n<\/strong><\/span>ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC=120\u00b0
\nCalculate:
\ni)\u00a0\u2220BEC
\nii)\u00a0\u2220 BED
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 20.
\n<\/strong><\/span>In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If angle ACO = 30\u00b0, find:
\n(i)\u00a0angle BCO
\n(ii)\u00a0angle AOB
\n(iii)\u00a0angle APB
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 21.<\/strong><\/span>
\nABC 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.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 22.
\n<\/strong><\/span>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 –
\ni)\u00a0\u2220ONL +\u00a0\u2220OML = 180\u00b0
\nii)\u00a0\u2220BAM =\u00a0\u2220BMA
\niii) ALOB is a cyclic quadrilateral.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 23.<\/strong><\/span>
\nThe given figure shows a semicircle with centre O and diameter PQ. If PA = AB and\u00a0\u2220BOQ = 140\u00b0; find measures of angles PAB and AQB. Also, show that AO is parallel to BQ.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 24.
\n<\/strong><\/span>The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20\u00b0\u00a0and angle POQ = 100\u00b0.
\nCalculate –
\ni) angle QTR
\nii) angle QRP
\niii) angle QRS
\niv) angle STR
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 25.
\n<\/strong><\/span>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:
\ni)\u00a0\u2220BAP =\u00a0\u2220ADQ
\nii)\u00a0\u2220AOB = 2\u2220ADQ
\n(iii)\u00a0\u2220ADQ =\u00a0\u2220ADB.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 26.<\/strong><\/span>
\nAB 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
\nSolution:<\/strong><\/span>
\n\"Selina
\n\"Selina<\/p>\n

Question 27.<\/strong><\/span>
\nTA 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 28.<\/strong><\/span>
\nTwo 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 29.<\/strong><\/span>
\nProve that any four vertices of a regular pentagon are concyclic (lie on the same circle)
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 30.<\/strong><\/span>
\nChords 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.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 31.<\/strong><\/span>
\nIn the given figure, find TP if AT = 16 cm and AB = 12 cm.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 32.
\n<\/strong><\/span>In the following figure, A circle is inscribed in the quadrilateral ABCD.
\n\"Selina
\nIf BC = 38 cm, QB = 27 cm, DC = 25 cm and that AD is perpendicular to DC, find the radius of the circle.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 33.<\/strong><\/span>
\nIn the figure, XY is the diameter of the circle, PQ is the tangent to the circle at Y. Given that\u00a0\u2220AXB = 50\u00b0 and\u00a0\u2220ABX = 70\u00b0. Calculate\u00a0\u2220BAY and \u2220APY.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 34.
\n<\/strong><\/span>In the given figure, QAP is the tangent at point A and PBD is a straight line. If\u00a0\u2220ACB = 36\u00b0 and\u00a0\u2220APB = 42\u00b0; find:
\ni)\u00a0\u2220BAP
\nii)\u00a0\u2220ABD
\niii)\u00a0\u2220QAD
\niv)\u00a0\u2220BCD
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 35.
\n<\/strong><\/span>In the given figure, AB is the diameter. The tangent at C meets AB produced at Q.
\n\"Selina
\nIf
\n\u2220CAB = 34\u00b0, find
\ni)\u00a0\u2220CBA
\nii)\u00a0\u2220CQB
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 36.
\n<\/strong><\/span>In the given figure, O is the centre of the circle. The tangets at B and D intersect each other at point P.
\n\"Selina
\nIf AB is parallel to CD and\u00a0\u2220ABC = 55\u00b0, find:
\ni)\u00a0\u2220BOD
\nii)\u00a0\u2220BPD
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 37.
\n<\/strong><\/span>In the figure given below PQ =QR,\u00a0\u2220RQP = 68\u00b0, PC and CQ are tangents to the circle with centre O. Calculate the values of:
\ni) \u2220QOP
\nii) \u2220QCP
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 38.<\/strong><\/span>
\nIn two concentric circles prove that all chords of the outer circle, which touch the inner circle, are of equal length.
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 39.
\n<\/strong><\/span>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.
\nGiven that AB = 8 cm, calculate PQ.
\n\"Selina
\nSolution:<\/strong><\/p>\n

\"Selina<\/p>\n

Question 40.<\/strong><\/span>
\nIn 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\u00a0\u2220XTY = 80\u00b0\u00a0and\u00a0\u2220XOZ = 140\u00b0, calculate the value of\u00a0\u2220ZXY.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 41.<\/strong><\/span><\/p>\n

In the given figure, AE and BC intersect each other at point D. If\u00a0\u2220CDE=90\u00b0, AB = 5 cm, BD = 4 cm and CD = 9 cm, find AE.
\n\"Selina
\nSolution:<\/strong><\/p>\n

\"Selina<\/p>\n

Question 42.<\/strong><\/span>
\nIn the given circle with centre O,\u00a0\u2220ABC = 100\u00b0,\u00a0\u2220ACD = 40\u00b0 and CT is a tangent to the circle at C. Find\u00a0\u2220ADC and\u00a0\u2220DCT.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

Question 43.<\/strong><\/span>
\nIn the figure given below, O is the centre of the circle and SP is a tangent. If\u00a0\u2220SRT = 65\u00b0, find the values of x, y and z.
\n\"Selina
\nSolution:<\/strong><\/span>
\n\"Selina<\/p>\n

More Resources for Selina Concise Class 10 ICSE Solutions<\/strong><\/p>\n