15 ircles-tangent roperties onstruction of tangent at a point on the circle. onstruction of tangents when the angle between radii is given. Tangents from an external point - construction and proof Touching circles - meaning, construction and proof. ommon tangents - DT and TT. onstruction of direct and transverse common tangents. The circle is the most primitive and rudimentary of all human inventions. It is the corner stone in the foundation of science and technology. It is the basic tool used by engineers, designers and the greatest artists and architects in the history of mankind. This unit facilitates you in, defining secant and tangent. differentiate between secant, chord and tangent. stating and verifying the tangent property. 'In a circle the radius at the point of contact is perpendicular to the tangent' stating the converse of the abo ve statement. explaining the meaning of touching circles-externally and internally. identifying the relation between d, R and r in touching circles. proving the theorems logically. solving numerical problems and riders based on theorems and their converse. constructing a tangent to a circle at a point on it. constructing tangents to a circle from an external point. stating the properties of DT and TT. constructing DT and TT to two given circles. measuring and verifying the length of DT and TT. The ancient Indian symbol of a circle, with a dot in the middle, known as 'bindu', was probably instrumental in the use of circle as a representation of the concept zero.
352 UNIT-15 In the previous chapter we have studied about chord properties of a circle. In this chapter let us study some more properties of circle. If a circle and a straight line are drawn in the same plane there are three possibilities as shown below. Figure 1 Figure 2 Figure 3 * Straight line R does not touch the circle. * R is a non-intersecting line with respect to the circle. * Straight line R intersects the circle at and. * R is called a secant of the circle. * Straight line R touches the circle at one and only one point. * R is called the tangent to the circle at. Secant: straight line which intersects a circle at two distinct points is called a Secant. Tangent: straight line which touches the circle at only one point is called a tangent. bserve the following figures. Tangents touch the circle at only one point. E R G S H D F, D, EF and GH are the tangents drawn to circles with centre.,, R and S are the points of contact or points of tangency. oint of contact: The point where a tangent touches the circle is called the point of contact. bserve the following examples. bjects in daily life situations which represent circles and their tangents are given.
ircle-tangent roperties 353 Example 1: Here is a pulley representing a circle. The parts of the string and D represent tangents to the pulley. Example 2: bserve the single wheel cycle on the floor. The wheel of the cycle represents a circle and the path on the floor where the cycle is moving represents a tangent to the wheel. roperties of tangents: Now let us learn an important property of tangents to a circle. ctivity 1 : bserve the given figures and complete the following table. L X M K D Y Fig. 1 Fig. 2 Fig. 3 N 1. Radius 2. Tangent 3. oint of contact 4. ngle between the radius and the tangent (measure the angle using protractor) Fig.1 Fig. 2 Fig. 3 X =... =... LMK =... From the above observations we can state that, In any circle, the radius drawn at the point of contact is perpendicular to the tangent. ctivity 2 : bserve the following figures, measure and write the angles in the given table. Fig. 1 Fig. 2 Fig. 3
354 UNIT-15 nswer the following questions : 1. In which figure radius is perpendicular to? 2. What is called in figure 3? 3. In which figure is a tangent? Why? From the activity 2, we can state that, In a circle, the perpendicular to the radius at its non-centre end is the tangent to the circle. This is the converse statement of tangent property stated earlier. onstruction of a tangent at a point on the circle. Example 1 : onstruct a tangent at any point on a circle of radius 3 cm. ngle Fig. 1 Fig. 2 Fig. 3 1.......... 2.......... Step 1 : With as centre draw a circle of radius 3 cm. mark a point on the circle, Join. Step 2 : With radius equal to and as centre draw an arc to intersect the circle at. 3 cm 3 cm Step 3 : Mark on the arc such that =. With the same radius, and as centre draw two intersecting arcs Step 4 : Join and produce on either sides to get the required tangent. 3 cm 3 cm Example 2 : Draw a circle of radius 2 cm and construct a pair of tangents at the non-centre end points of two radii such that the angle between the radii is 100. Measure the angle between the tangents. Given: Radius of the circle = r = 2 cm; ngle between the radii = 100
ircle-tangent roperties 355 Step 1 : With as centre draw Step 2 : onstruct a tangent at. a circle of radius 2 cm. Mark on it and join. 2 cm 2 cm Step 3 : t draw =100 Step 4 : t construct another tangent. Measure angle between the tangents. 80 2 cm 2 cm ngle between the tangents is 80º. bserve that, in a circle angle between the radii and angle between the tangents drawn at their non-centre ends are supplementary.
356 UNIT-15 Example 3: Draw a chord of length 4 cm in a circle of radius 2.5cm. onstruct tangents at the ends of the chord. Given: r = 2.5cm, length of the chord = 4cm Step 1 : With as centre draw a circle Step 2 : onstruct a chord of length of radius 2.5 cm. 4cm. 4 cm Step 3 :Join and. Step 4 : onstruct tangents at and to the circle. 4 cm 4 cm
ircle-tangent roperties 357 orollaries : 1. The perpendicular to the tangent at the point of contact passes through the centre of the circle. 2. nly one tangent can be drawn to a circle at any point on it. 3. Tangents drawn at the ends of a diameter are parallel to each other. Discuss the reasons for the above three statements in the class and explain them. EXERISE 15.1 1. Draw a circle of radius 4 cm and construct a tangent at any point on the circle. 2. Draw a circle of diameter 7 cm and construct tangents at the ends of a diameter. 3. In a circle of radius 3.5cm draw two mutually perpendicular diameters. onstruct tangents at the ends of the diameters. 4. In a circle of radius 4.5 cm draw two radii such that the angle between them is 70. onstruct tangents at the non-centre ends of the radii. 5. Draw a circle of radius 3 cm and construct a pair of tangents such that the angle between them is 40. 6. Draw a circle of radius 4.5 cm and a chord of length 7cm in it. onstruct a tangent at. 7. In a circle of radius 5 cm draw a chord of length 8 cm. onstruct tangents at the ends of the chord. 8. Draw a circle of radius 4cm and construct chord of length 6 cm in it. Draw a perpendicular radius to the chord from the centre. onstruct tangents at the ends of the chord and the radius. 9. Draw a circle of radius 3.5 cm and construct a central angle of measure 80 and an inscribed angle subtended by the same arc. onstruct tangents at the points on the circle. Extend tangents to intersect. What do you observe? 10. In a circle of radius 4.5cm draw two equal chords of length 5cm on either sides of the centre. Draw tangents at the end points of the chords.
358 UNIT-15 Tangents to a circle from a point at different positions. 1. oint on the circle ctivity 1 : Draw a circle of radius 3 cm and mark a point on it. onstruct a tangent to it at. Try to construct another tangent at. Is it possible? Why? You have already learnt that only one tangent can be drawn to the circle at a point on it. 2. oint inside the circle. ctivity 2: Draw a circle of radius 3cm. Mark a point inside the circle. Is it possible to draw a tangent to the circle from this point? You will find that each line drawn through this point intersects the circle at two distinct points. ll these straight lines are secants to the circle. 3. oint outside the circle ctivity 3: Draw a circle of radius 3cm. Mark a point outside the circle. Draw rays from this point towards the circle. bserve them, how many secants can be drawn to the circle from? How many tangents can be drawn to the circle from the external point? From the figure, you observe that only and R touches the circle. They are tangents to the circle drawn from. nly two tangents can be drawn from an external point to a circle. The two tangents that can be drawn from an external point to a circle have special properties. You can discover these properties by the following activity. bserve the figures given below and answer the following questions. X. H F Y D G E Fig. 1 Fig. 2
ircle-tangent roperties 359 (a) Name the tangents drawn from the external point to the circle. (b) Measure and. What is your conclusion? (c) Measure and by using protractor. (d) Measure and. Where are the angles formed? Record the measurements in the table given. Figure I II III 1. =... =... =... =... =... =... 2. =... =... =... =... =... =... fter observing the data complete the following statements. 1. The tangents drawn from an external point to a circle are... 2. The tangents drawn from an external point to a circle subtend... angles at the centre. 3. The tangents drawn from an external point to a circle are... inclined to the line joining the centre and an external point. From the above activity you find that tangents drawn from an external point to a circle (a) are equal (b) subtend equal angles at the centre. (c) are equally inclined to the line joining the centre and the external point. We have learnt the property of tangents drawn from an external point to a circle by practical construction and measurement. Now, let us discuss the logical proof for this property. ut, before going to logical proof, answer the following questions. Dicuss in the class. This will help you to prove the theorem. 1. What type of angles are and? 2. What are and with respect to the circle? 3. What type of triangles are and? 4. What is with respect to and? 5. re and congruent to each other? 6. ccording to which postulate the triangles and are congruent? Now let us prove the theorem logically.
360 UNIT-15 Theorem: The tangents drawn from an external point to a circle (a) are equal (b) subtend equal angles at the centre (c) are equally inclined to the line joining the centre and the external point Data : is the centre of the circle. is an external point. and are the tangents., and are joined. To rove: (a) = (b) = (c) = roof : Statement Reason In and = = = 90 hyp = hyp radii of the same circle Radius drawn at the point of contact is perpendicular to the tangent. ommon side RHS Theorem (a) = T (b) = (c) = ED ILLUSTRTIVE EXMLES Example 1 : In the figure, a circle is inscribed in a quadrilateral D in which = 90. If D = 23cm, = 29cm and DS = 5cm find the radius of the circle. Sol: In the figure.,, D and D are the tangents drawn to the circle at,, S and R respectively. DS = DR (tangents drawn from an external point D to the circle.) but DS = 5cm (given) DR = 5 cm In the fig. D = 23 cm, but (given) R = D DR = 23 5 = 18 cm R = (tangents drawn from an external point to the circle) = 18 cm. D R S r r
ircle-tangent roperties 361 If = 18 cm then (given = 29 cm) = = 29 18 = 11 cm In quadrilateral, = (tangents drawn from an external point ) = = = 90 (given) (radii of the same circle) = = 90 (angle between the radius and the tangent at the point of contact.) is a square. radius of the circle, = 11 cm Example 2 : In the fig. and are the tangents drawn from an external point. rove that and are supplementary. Sol: Data : is the centre of the circle. is an external point. and are the tangents drawn from an external point. To prove : + = 180 roof : In quadrilateral, + + + = 360 but = 90 0 and = 90 + 90 0 + + 90 0 = 360 + + 180 0 = 360 + = 360 180 + = 180 0 and are supplementary. Example 3 : In the figure, and R are the tangents to a circle with centre. If = 4. Find and. 5 Sol: In the figure R + R = 180 ( pposite angles are supplementary) but, R = 4 5 R 4 R + R = 180 5 R 4 R + 5 R = 180 5 9 R = 5 180 R= 5 180 0 0 100 9 R = 100 If R = 100 then R = 180 R = 180 100 = 80 R = 80
362 UNIT-15 Example 4 : In the figure is the centre of the circle. The tangents at and D intersect each other at point. If is parallel to D and = 55 Find (i) D (ii) D. Sol: In the figure D (given) = D ( alternate angles) but = 55 ( given) D = 55 In the figure D = 2 D D = 2 55 = 110. but D + D = 180 (central angle is twice the inscribed angle.) D = 180 D = 180 110 = 70 Example 5 : In the figure, show that perimeter of = 2 ( + + R). Sol: erimeter of = + + = + + + + R + R but = R (tangents drawn from to the circle) = = R (tangents drawn from to the circle) (tangents drawn from to the circle) erimeter of = + + + R + R + = 2 + 2 + 2R = 2( + + R) D R EXERISE 15.2. Numerical problems based on tangent properties. 1. In the figure, R and are the tangents to the circle. touches the circle at X. If = 7cm, find the perimeter of. 2. In the figure, if R = 50 find the measure of SR. S X R 3. Two concentric circles of radii 13cm and 5cm are drawn. Find the length of the chord of the outer circle which touches the inner circle. R 4. In the given, = 12cm, = 8 cm and = 10cm. Find F, D and E. F D E
ircle-tangent roperties 363 5. is a right angled triangle, right angled at. circle is inscribed in it. The lengths of the sides containing the right angle are 12cm and 5cm. Find the radius of the circle. D 6. In the given quadrilateral D, = 38cm, = 27cm, S D = 25cm and D D find the radius of the circle. R 7. In the given figure =, = 68 0 D and D are the tangents to the circle with centre. alculate the measure of (i) (ii) and (iii) D. Riders based on tangent properties. 1. quadrilateral D is drawn to circumscribe a circle. rove that +D=D+. 2. Tangents and are drawn to circle with centre, from an external point. rove that = 2.. D 3. In the figure two circles touch each other externally at. is a direct common tangent to these circles. rove that (a) tangent at bisects at (b) = 90. 4. pair of perpendicular tangents are drawn to a circle from an external point. rove that length of each tangent is equal to the radius of the circle. 5. If the sides of a parallelogram touch a circle. rove that the parallelogram is a rhombus. 6. In the figure, if = prove that =. R 2 7. ircles 1 and 2 touch internally at a point and is a chord of the circle 1 intersecting 2 at. rove that =. I 1 8. circle is touching the side of at. and when produced are touching the circle at and R respectively. rove that = ½ (perimeter of )
364 UNIT-15 onstruction of tangents from an external point to a circle. Example: Draw a circle of radius 1.5cm. onstruct two tangents from a point 2.5cm away from the centre. Measure and verify the length of the tangent. Given r = 1.5 cm, d = 2.5 cm Step 1: Draw a line segment equal to 2.5 cm. Draw a circle of radius 1.5 cm with as centre. Step 2:Draw the pe rpendicu lar bisector of and mark the mid point of as M. M Step 3: With M or M as radius draw a circle to intersect the circle already drawn mark and name the intersecting points as and. Step 4:Join and. and are the required tangents. M M Step 5: Length of the tangent by measurement = = 2 cm Verification by calculation. t = 2 2 d r = 2 2 2.5 1.5 6.25 2.25 4 2cm EXERISE 15.3 1. Draw a circle of radius 6 cm and construct tangents to it from an external point 10 cm away from the centre. Measure and verify the length of the tangents. 2. onstruct a pair of tangents to a circle of radius 3.5cm from a point 3.5cm away from the circle. 3. onstruct a tangent to a circle of radius 5.5cm from a point 3.5 cm away from it. 4. Draw a pair of perpendicular tangents of length 5cm to a circle. 5. onstruct tangents to two concentric circles of radii 2cm and 4cm from a point 8cm away from the centre.
ircle-tangent roperties 365 Touching ircles bserve the pairs of circles given below. D E F Fig. 1 Fig. 2 Fig. 3 In fig. 1, two circles with centres and are away from each other. They do not have any common points, In fig. 2, two circles with centres and D intersect at and. They have two common points, and. In fig. 3, two circles with centres E and F have a common point. They are called touching circles. Touching circles: Two circles having only one common point of contact are called touching circles. 1 2 2 1 1 Fig. 5 Fig. 6 In fig. 4 two circles with centres and 1 have no common point. They have different centres. In figure: 5 two circles 1 and 2 have a common centre. ut they do not have a common point on the circle. Such circles are called concentric circles. In figure 6, circles 1 and 2 with centres and have a common point between them. These are also touching circles. Touching circles : Two circles having only one common point of contact are called touching circles. Two circles, one out side the other and having a common point of contact are called externally touching circles (or) If the centres of the two touching circles lie on either sides of the point of contact, then they are called externally toucing circles. Two circles one inside the other and having a common point of contact are called internally touching circles (or) If the centres of the circles lie on the same side of the point of contact, then they are called internally touching circles. External touching circles Internal touching circles
366 UNIT-15 Relation between the distance between centres of touching circles and their radii. ctivity 1: bserve the following figures. Measure and enter the data in the table given below. Figure 1 Figure 2 Fig. (R) (r) (d) Relation between, and i.e., R, r and d 1 2 From the above observations, you find that (i) If two circles touch each other externally, the distance between their centres is equal to the sum of their radii [d = R + r] (ii) If two circles touch each other internally, the distance between their centres is equal to the difference of their radii. [d = R r] ctivity 2: Draw two circles of radii 5cm and 3cm whose centres are at 8cm apart. What relation do you find between sum of the radii and distance between their centres? ctivity 3: Draw two circles of radii 5cm and 3cm whose centres are at 2cm apart. What relation do you find between difference of the radii and the distance between their centres. roperty related to the centres and the point of contact of touching circles. With reference to the above touching circles, 1. Name (a) the point of contact. (b) their centres. 2. What type of angle is in figure 1 and in figure 2? 3. How are the points,, lying in the figure 1 and,, in figure 2? Why? From the above acitivity we can conclude that, If two circles touch each other, their centres and the point of contact are collinear. Now let us prove the theorem logically.
ircle-tangent roperties 367 Theorem : If two circles touch each other, the centres and the point of contact are collinear. ase 1: If two circles touch each other externally, the centres and the point of contact are collinear. X Data : and are the centres of touching circles. is the point of contact. To prove :, and are collinear onstruction : Draw the tangent XY. roof : Statement Reason In the figure } X = 90...(1) Radius drawn at the point of contact X = 90...(2) is perpendicular to the tangent. Y X + X = 90 + 90 by adding (1) and (2) = 180 is a straight line, and are collinear. ED is a straight angle ase 2 : If two circles touch each other internally the centres and the point of contact are collinear. Data To prove : and are the centres of touching circles. is the point of contact. :, and collinear onstruction : Draw the tangent XY. Join and. roof : Statement Reason In the figure, X = 90... (1) Radius drawn at the point of contact X = 90... (2) is perpendicular to the tangent. X = X = 90 } and lie on the same line., and are collinear. ED X Y
368 UNIT-15 ILLUSTRTIVE EXMLES Example 1.Three circles touch each other externally. Find the radii of the circles if the sides of the triangle obtianed by joining the centres are 10cm, 14cm and 16cm respectively. Sol. ircles with centres, and touch each other externally at, and R respectively as shown in the fig. Let = x = = 10 x R = = 14 x but, + = 16 14 x + 10 x = 16 x x R 24 2x = 16 24 16 = 2x 2x = 8 x = 8 2 = 4 R = = 4cm = 10 4 = 6cm R = 14 4 = 10cm radius of the circle with centre radius of the circle with centre radius of the circle with centre Example 2.In the figure and are the centres of the circles with radii 9cm and 2cm respectively. If R = 90 and = 17cm find the radius of the circle with centre R. Sol. Let the radius of the circle with centre R = x units. In R, = 17cm R = (x + 9) cm R = (x + 2) cm and R = 90 2 = R 2 + R 2 (ythagoras theorem) 17 2 = (x + 9) 2 + (x + 2) 2 289 = x 2 + 81 + 18x + x 2 + 4 + 4x 2x 2 + 22x + 85 289 = 0 2x 2 + 22x 204 = 0 by 2 x 2 + 11x 102 = 0 (x + 17)(x 6) = 0 x + 17 = 0 or x 6 = 0 x = 17 or x = 6 radius of the circle with centre R = 6cm R x x 2 9
ircle-tangent roperties 369 Example 3. In the figure circles with centres and touch each other internally. is the point of contact. rove that R. Sol. Let In, Similarly, In R, from (1) and (2) = xº. = (radii of the same circle) = (angles opposite to equal sides of an isosceles le) = x...(1) ( = x ) = R (radii of the same circle) R = R (angles opposite to equal sides of an isosceles le) R = xº...(2) ( R = x = R corresponding angles are equal. R Hence proved. EXERISE 15.4. Numerical problems on touching circles. R (angles which are equal to same angle are equal) 1. Three circles touch each other externally. Find the radii of the circles if the sides of the triangle formed by joining the centres are 7cm, 8cm and 9cm respectively. 2. Three circles with centres, and touch each other as shown in the figure. If the radii of these circles are 8 cm, 3 cm and 2 cm respectively, find the perimeter of. 3. In the figure = 10 cm, = 6 cm and the radius of the smaller circle is x cm. Find x. Riders based on touching circles. 1. straight line drawn through the point of contact of two circles with centres and intersect the circles at and respectively. Show that and are parallel. 2. Two circles with centres X and Y touch each other externally at. Two diameters and D are drawn one in each circle parallel to other. rove that, and are collinear. 3. In circle with centre, diameter and a chord D are drawn. nother circle is drawn with as diameter to cut D at. prove that D = 2. x
370 UNIT-15 4. In the given figure = 8cm, M is the mid point of. Semicircles are drawn on with M and M as diameters. circle with centre '' touches all three semicircles as shown. rove that the radius of this circle is 1 6. onstruction of touching circles. Example 1: Draw two circles of radii 2cm and 1cm to touch each other internally and mark the point of contact. Given : 1 = R = 2cm, 2 = r = 1cm Step 1: Find 'd' d = R r = 2 1 = 1cm Step 2: Draw = 1cm, with as centre and radius equal to 2cm draw a circle. Step 3: With as centre and radius equal to 1cm draw a circle. Step 4: roduce to meet the point of contact. EXERISE 15.5 1. Draw two circles of radii 5 cm and 2 cm touching externally. 2. onstruct two circles of radii 4.5 cm and 2.5 cm whose centres are at 7 cm apart. 3. Draw two circles of radii 4 cm and 2.5 cm touching internally. Measure and verify the distance between their centres. 4. Distance between the centres of two circles touching internally is 2 cm. If the radius of one of the cirles is 4.8 cm, find the radius of the other circle and hence draw the touching circles. Study the following examples. ommon tangents Example 1 : Look at the wheels of the bicycle representing the circles, which moves along the road representing a line. The line touches both the wheels. Example 2 : bserve the following figures, in each case you have two circles touching the same straight line. The straight line is a tangent to both circles. Such a tangent is called a common tangent. 1 2 x R M D E D F G H bserve the position of the circles with reference to the common tangents. You find that both the circles lie on the same side of the tangent.
ircle-tangent roperties 371 If both the circles lie on the same side of a common tangent, then the common tangent is called a direct common tangent (DT). Example 3 : bserve the following figures. In each case the straight line is a common tangent. ut observe the position of the circles. The circles lie on either side of the common tangent. D E F G H If both the circles lie on either side of a common tangent, then the common tangent is called a transverse common tangent (TT). ctivity: bserve the figures, in each case identify the types and number of common tangents.' enumbra Sun Earth E E D D Umbra onstruction of direct common tangents to two congruent circles 1. Draw direct common tangets to two circles of radii 3cm, whose centres are 8cm apart. (construction shown here is as per suitable scale) Given: r = 3cm = 1 = 2, d = 8cm Step 1: Draw = 8cm, wtih and as centres draw two circles of radii 3cm. 1 2 Step 2: Draw two perpendicular diameters at and using compasses. Name the points as,, R and S. R S
372 UNIT-15 Step 3: Join and RS. and RS are the required direct common tangets. Note: R 1. Method of constructing DT's is same for congruent (a) Non-intersecting circles (d > R + r) (b) ircles touching externally (d = R + r) and (c) Intersecting circles (d < R + r and d > R r) 2. Direct common tangents drawn to two congruent circles are equal and parallel to the line joining the centres. S To construct a direct common tangent to two non-congruent circles Example 1 : onstruct a direct common tangent to two circles of radii 5cm and 2cm whose centres are 10cm apart. Measure and verify the length of the direct common tangent. (construction done according to suitable scale) Given:d = 10cm, 1 = R = 5cm, 2 = r = 2cm, 3 = R r = 5 2 = 3cm Step 1:Draw = 10cm, with and as centres and radius 5cm and 2cm respectively draw circles 1 and 2. Step 2 :With as centre and radius 3cm draw 3. 1 2 Step 3: Draw bisector to intersect at M. with M as centre and M as radius draw 4 to intersect 3 at and. M 2 3 1 3 1 2 Step 4: Join and produce it to meet 1 at S. S M 2 3 1 4 4
ircle-tangent roperties 373 Step 5:Draw tangent. S M 2 3 1 4 Note : Similarly another DT can be constructed to 1 and 2. Step 7: y measurement, ST = 9.5 cm Step 6:With S as centre and radius draw an arc to intersect 2 at T. Join ST and T. S M 2 3 1 4 Tips : To check the accuracy of construction, TS must be a rectangle. T 2 2 2 2 y calculation, DT = d (R r) = 10 (5 2) = 100 9 = 91 = 9.5cm onstruction of transverse common tangents to two circles Example 1 : onstruct a transverse common tangent to two circles of radii 4cm and 2cm whose centres are 10cm apart. Measure and verify the length of the transverse common tangent. (constructions done according to suitable scale) Step 1: d = 10 cm, 1 = R = 4 cm, 2 = r = 2 cm, 3 = R + r = 4 + 2= 6cm Step 2 : Draw = 10cm with and as centres draw circles of radii 4 cm and 2 cm respectively. Name the circle as 1 and 2. Step 3 : With as centre and radius 6cm, draw 3. 1 2 Step 4:Draw a bisector to intersect at M. With M as centre and M as radius draw 4 to cut 3 at and. 3 1 2 Step 5:Join to intersect 1 at S. 4 4 M 2 S M 2 1 1 3 3
374 UNIT-15 Step 6 :Draw tangent. Step 7:With S as centre and radius draw an arc to intersect 2 at T. Join ST. 4 4 3 1 S M 2 Note : Similarly another TT can be constructed to 1 and 2. 3 1 S M T 2 Step 8 : y measurement, ST = 8cm, 2 2 2 2 y calculation, TT = d (R r) = 10 (4 2) = 100 36 = 64 = 8cm Note: (1) Method of constructing DT's is same for non congruent. (a) Non-intersecting circles (d > R + r) (b) ircles touching externally (d = R + r) and (c) Intersecting circles (d < R + r) Method of constructing TT's is same for congruent and non-congruent non-intersecting circles (d > R + r) EXERISE 15.6 I. (). 1. Draw two congruent circles of radii 3 cm, having their centres 10 cm apart. Draw a direct common tangent. 2. Draw two direct common tangents to two congruent circles of radii 3.5 and whose distance between them is 3 cm. 3. onstruct a direct common tangent to two externally touching circles of radii 4.5 cm. 4. Draw a pair of direct common tangents to two circles of radii 2.5 cm whose centres are at 4 cm apart. (). 1. onstruct a direct common tangent to two circles of radii 5 cm and 2cm whose centres are 3 cm apart. 2. Draw a direct common tangent to two internally touching circles of radii 4.5 cm and 2.5 cm. 3. onstruct a direct common tangent to two circles of radii 4 cm and 2 cm whose centres are 8 cm apart. Measure and verify the length of the tangent. 4. Two circles of radii 5.5 cm and 3.5 cm touch each other externally. Draw a direct common tangent and measure its length. 5. Draw direct common tangents to two circles of radii 5 cm and 3 cm having their centres 5 cm apart.
ircle-tangent roperties 375 6. Two circles of radii 6 cm and 3 cm are at a distance of 1 cm. Draw a direct common tangent, measure and verify its length. II.(). 1. Draw a transverse common tangent to two circles of radii 6 cm and 2 cm whose centres are 8 cm apart. 2. Two circles of radii 4.5 cm and 2.5 cm touch each other externally. Draw a transverse common tangent. 3. Two circles of radii 3 cm each have their centres 6cm apart. Draw a transverse common tangent. (). 1. Draw a transverse common tangent to two congruent circles of radii 2.5 cm, whose centres are at 8 cm apart. 2. Two circles of radii 3 cm and 2 cm are at a distance of 3 cm. Draw a transverse common tangent and measure its length. 3. Draw transverse common tangents of length 8 cm to two circles of radii 4 cm and 2 cm. 4. onstruct a transverse common tangent to two circles of radii 4 cm and 3 cm whose centres are 10 cm apart. Measure and verify by calculation. 5. onstruct two circles of radii 2.5 cm and 3.5 cm whose centres are 8 cm apart. onstruct a transverse common tangent. Measure its length and verify by calculation. Tangent roperties onstructions roperties ommon tangents to two circles. Tangent at a point on a circle DT TT Tangents at the non-centre ends of the radii To two congruent circles To two noncongruent circles Externally non-touching Externally touching Tangent to a circle if angle between the tangents is given Tangent at the ends of the chord Tangent at the ends of the diameter * Externally non-touching * Externally touching * Intersecting Tangents from an external point to circle Internally touching Touching circles Numerical roblems Riders EXERISE 15.2 NSWERS 0. 1] 14 cm 2] SR 65 3] = 24cm 4] F = 7cm, D = 5 cm, E = 3 cm 5] 2cm 6] 14cm 7] (i) 0 56 (ii) 0 112 (iii) 0 D 68 EXERISE 15.4. 1] 4cm, 3cm and 5cm 2] 16cm 3] x = 2.4 cm
376 UNIT-15 Let us summarise the data regarding common tangents. Study the given table. Sl. ircles Figures Relation between Number of Number of Number of No. d, R and r DT'S TT'S T'S 1. Non-intersecting d > R + r 2, 2, 4 and non-touching, RS TU, VW circles 2. ircles touching d = R + r 2, 1, 3 externally, RS XY 3. Intersecting circles d < R + r 2, - 2 and d > R r MN, 4. ircles touching d = R r 1, - 1 internally EF 5. Non-concentric d < R r - - - circles and d > 0 6. oncentric circles d = 0 - - -