{"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 Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 18 Tangents and Intersecting Chords<\/strong><\/p>\n Question 1.<\/strong><\/span> 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. <\/p>\n Question 3.<\/strong><\/span> Question 4.<\/strong><\/span> Question 5.<\/strong><\/span> Question 6.<\/strong><\/span> Question 7.<\/strong><\/span> Question 8.<\/strong><\/span> From A, AP and AS are tangents to the circle. Question 9. Question 10. <\/p>\n Question 11.<\/strong><\/span> Question 12. Question 13. Question 14. Question 15.<\/strong><\/span> Question 16.<\/strong><\/span> Question 17. Question 18. Question 19.<\/strong><\/span> Question 20.<\/strong><\/span> Question 21. Question 22. Question 23.<\/strong><\/span> Question 24. Now, Question 1.<\/strong><\/span> Question 2.<\/strong><\/span> Question 3.<\/strong><\/span> Question 4.<\/strong><\/span> Question 5.<\/strong><\/span> Question 6.<\/strong><\/span> Question 7.<\/strong><\/span> Question 8.<\/strong><\/span> Question 9.<\/strong><\/span> Question 10.<\/strong><\/span> Question 11.<\/strong><\/span> Question 12.<\/strong><\/span> Question 13.<\/strong><\/span> Question 14.<\/strong><\/span> Question 15.<\/strong><\/span> Question 16.<\/strong><\/span> Question 1.<\/strong><\/span> Question 2. Question 3.<\/strong><\/span> Question 4.<\/strong><\/span> Question 5.<\/strong><\/span> Question 6.<\/strong><\/span> Question 7. Question 8.<\/strong><\/span> ABCD is a cyclic quadrilateral. \u2235 \u2220A = 3\u2220C \u2234 \u2220B+ \u2220D = 180\u00b0 \u2235\u2220D = 5\u2220B Question 9.<\/strong><\/span> Question 10.<\/strong><\/span>Tangents and Intersecting Chords Exercise 18A –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n
\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<\/p>\n
\n
\nSolution:<\/strong><\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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<\/p>\n
\nIf the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.
\n
\nSolution:<\/strong><\/span>
\n<\/p>\n
\nIf the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.
\n
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\n<\/strong><\/span>From the given figure prove that:
\nAP + BQ + CR = BP + CQ + AR.
\n
\nAlso, show that AP + BQ + CR = \\(\\frac{1}{2}\\)\u00a0\u00a0\u00d7\u00a0perimeter of triangle ABC.
\nSolution:<\/strong><\/span>
\n<\/p>\n
\n<\/strong><\/span>In the figure, if AB = AC then prove that BQ = CQ.
\n
\nSolution:<\/strong><\/p>\n
\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<\/p>\n
\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<\/p>\n
\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
\ni) tangent at point P bisects AB.
\nii) Angle APB = 90\u00b0
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\n<\/p>\n
\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
\nCalculate the value of x, the radius of the inscribed circle.
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\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
\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>\nTangents and Intersecting Chords Exercise 18B –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n
\ni) In the given figure, 3 x CP = PD = 9 cm and AP = 4.5 cm. Find BP.
\n
\nii) In the given figure, 5 x PA = 3 x AB = 30 cm and PC = 4cm. Find CD.
\n
\niii) In the given figure, tangent PT = 12.5 cm and PA = 10 cm; find AB.
\n
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\nIf PQ is a tangent to the circle at R; calculate:
\ni) \u2220PRS
\nii) \u2220ROT
\n
\nGiven: O is the centre of the circle and \u2220TRQ = 30\u00b0
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\nTangents and Intersecting Chords Exercise 18C –\u00a0Selina Concise Mathematics Class 10 ICSE Solutions<\/h3>\n
\nProve that of any two chord of a circle, the greater chord is nearer to the centre.
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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
\n<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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<\/p>\n
\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
\nSolution:<\/strong><\/span>
\n<\/p>\n
\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<\/p>\n
\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
\n\u21d2 \u2220A = 3 \u00d7 45\u00b0
\n\u21d2 \u2220A = 135\u00b0
\nSimilarly,<\/p>\n
\n\u21d2\u2220B + 5\u2220B = 180\u00b0
\n\u21d2 6\u2220B = 180\u00b0
\n\u21d2 \u2220B = 30\u00b0<\/p>\n
\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
\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<\/p>\n
\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>