In the adjoining figure circles with centres X and Y touch each other at point Z. A secant passing
Construction: Draw segments XZ and YZProof: By theorem of touching circles, points X, Z, Y are collinear.XZA BZY vertically opposite anglesLet XZA = BZY = a ….. (I)Now, seg XA seg XZ …….. (Radii of same circle)XAZ = XZA. = a …….. (isosceles triangle theorem) (II)similarly, seg YB seg YZ (Radii of same circle)BZY = […]
In figure 3.86, circle with centre M touches the circle with centre N at point T.
(1) Given radius of bigger circle is 9cm.Length of segment MT = 9cm.(2) MT = MN+NT [M-N-T]9 = MN+2.5 [Given radius of smaller circle is 2.5]MN = 9-2.5MN = 6.5cm.(3) RM touches smaller circle at S. MR is the tangent to the smaller circle. NS is the radius of smaller circle.NSM = 90˚ [Tangent theorem]In […]
In figure 3.85, ABCD is a parallelogram. It circumscribes the circle with centre T.
Given ABCD is a parallelogram.AB = DC ……(i) [Opposite sides of parallelogram are equal]AD = BC ………(ii)AE = AH………(iii) [Two tangents from a common point are congruent]BE = BF……….(iv)CG = CF………..(v)DG = DH………..(vi)Adding (iii), (iv), (v),(vi)AE+BE+CG+DG = AH+BF+CF+DHAB+CD = AD+BC ……(vii) [AH+DH = AD, BF+CF = BC]From (i), (ii) and (vii)2AB = 2ADAB = ADAD […]
In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle
Construction: Draw OC and OB. Proof:Given AB and AC are the tangents and r is the radius of the circle.AB = AC …….(i) [Two tangents from a common point are congruent]OB = OC = r …(ii) [radii of same circle]Given AB = rAB = AC =OB = OC [From (i) and (ii)]OBA = OCA = […]
In figure 3.83, M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = 6√3 then find –
(1) Given KL is the tangent. ML is the radius.MLK = 90˚ [Tangent theorem]MK = 12, KL = 6√3In MLK, ML2+KL2 = MK2 [Pythagoras theorem]ML2 = 122-(6√3)2 = 144-108 = 36ML = 6Hence radius of the circle is 6cm.(2) ML = ½ MKK = 30˚ [Converse of 30˚-60˚-90˚ theorem]M = 180-(90+30) = 60˚ [Angle sum property]Hence K = 30˚ […]
Line l touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following.
(1) Given radius = 9cm. Line l is the tangent to the circle.d(O,P) = 9cmOP is the radius of circle.(2) d(O,Q) = 8cm.d(O,Q) is less than radius. So Q will lie inside circle.(3) Point R can be on two locations on line l. d(O,R ) = 15InOPR, OPR = 90˚ [Tangent theorem]OP2+PR2 = OR2 [Pythagoras theorem]92+PR2 = 152PR2 = 152-92PR2 = 225-81 = […]
Four alternative answers for each of the following questions are given. Choose the correct alternative.
(1) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers ?(A) 4.4 cm (B) 8.8 cm (C) 2.2 cm (D) 8.8 or 2.2 cmSolution:If the circles touch each other externally, distance between their centres is equal to the sum of their radii.Distance between the […]
In figure 3.79, O is the centre of the circle and B is a point of contact. seg OE seg AD, AB = 12, AC = 8, find
(1) Given AB = 12 , AC = 8AB is the tangent . AD is the secant.AC×AD = AB2 [Tangent secant theorem]8×AD = 122AD = 144/8 = 18Hence measure of AD is 18 units(2) AD = AC+DC [A-C-D]18 = 8+DCDC = 18-8DC = 10Hence measure of DC is 10 units.(3) Given OEADOECD [A-C-D]DE = ½ CD […]
In figure 3.78, chord MN and chord RS intersect at point D.
(1) Given RD = 15, DS = 4, MD = 8RD×DS = MD×DN [Theorem of chords intersecting inside the circle]15×4 = 8×DNDN = 15×4/8 = 60/8DN = 7.5 units(2) RS = 18, MD = 9, DN = 8RD×DS = MD×DN [Theorem of chords intersecting inside the circle](RS-DS)×DS = MD×DN [RD +DS = RS](18-DS)DS = 9×818DS […]
In figure 3.77, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS.
Given PQ = 12 , PR = 8PQ is a tangent to the circle.PQ2 = PR ×PS122 = 8×PSPS = 144/8PS = 18PS = PR+RSRS = PS-PRRS = 18-8 = 10Hence PS = 18 units and RS = 10 units.