Circle-Chord properties

Transcription

1 14 ircle-hord properties onstruction of a chord of given length. Equal chords are equidistant from the centre. ngles in a segment. ongrue nt circles and concentric circles. onstruction of congruent and concentric circles. This unit facilitates you in, defining chord, segment and angle in a segment. construction of a chord of given length in a circle. stating and verifying that (a) equal chords are equidistant from the centre. (b) angles in the same segment are equal. (c) angles in the major segment are acute angles. (d) angles in the minor segment are obtuse angles. (e) angles in a semi-circle are right angles. differentiating congruent circles and concentric circles. constructing co ngruent circles and concentric circles. The circle has been known since before the beginning of recorded history. It is the basis of wheel, which with related inventions like gears, makes much of modern civilization possible. There is nothing strange in the circle being the origin of any and every marvel. - ristotle In mathematics, the study of circles has helped in the development of geometry and calculus.

2 344 UNIT-14 We have already learnt about the terms related to circle. bserve the figure and the table given below. ecall and identify the terms. Term 1. adius 2. hord Meaning Line segment joining the centre and a point on the circle. adius is denoted by r. The line segment joining any two points on the circle Name,, E, 3. iameter chord which passes through the centre of the circle. It is denoted by d. 4. rc It is a part of the circle. XE, E, 5. Segment (a) Minor segment (b) Major segment (c) Semi circular region The region bounded by an arc and a chord. egion bounded by a minor arc and a chord. egion bounded by a major arc and a chord. egion bounded by a semi - circle and diameter. XE,, X XE (egion bounded by minor arc XE and chord E) E (egion bounded by major arc E and chord E) X (egion bounded by semi-circle X and diameter )

3 ircle-hord roperties 345 onstruction of a chord of given length Example 1: raw a chord of length 5 cm in a circle of radius 3 cm. Sol: Step 1: With centre '', draw a circle of radius 3 cm. Mark a point '' on the circle. Step 2 : With as centre and radius equal to 5 cm draw an arc to cut the circle at. Join and. is the required chord of length 5 cm. 5 cm Example 2 : In a circle of radius 2.5 cm, construct a chord of length 4cm. Measure the distance between the centre and the chord. Step 1:With as centre, draw a circle of radius 2.5 cm and construct a chord LM = 4 cm. L 4 cm M Step 2 : Mark the mid point of the chord LM, say. Join and measure it. istance between the centre '' an d th e chord LM is = 1.5 cm. L 1.5 cm 4 cm M

4 346 UNIT-14 EXEISE raw a circle of radius 3.5 cm and construct a chord of length 6 cm in it. Measure the distance between the centre and the chord. 2. onstruct two chords of length 6 cm and 8 cm on the same side of the centre of a circle of radius 4.5cm. Measure the distance between the centre and the chords. 3. onstruct two chords of length 6.5cm each on either side of the centre of a circle of radius 4.5 cm. Measure the distance between the centre and the chords. 4. onstruct two chords of length 9cm and 7 cm on either side of the centre of a circle of radius 5 cm. Measure the distance between the centre and the chords. roperty related to the chords of a circle bserve the following circles. In each case, measure the lengths of the chords and find the distance between the chords and the centre. Figure 1 Figure 2 Figure 3 G E S H F M I V T N J L U K hords hords hords = = EF = GH = IJ = KL = MN = erpendicular distance erpendicular distance erpendicular distance = = = S = T = U = V = In each case compare the length of the chords and then their perpendicular distance from the centre. What is your conclusion? Equal chords of a circle are equidistant from the centre. ny number of chords can be drawn in a circle. If they are equal in length, they will be at equal distance from the centre. The converse of this statement is also true. If the chords of a circle are at equal distance from the centre, then they are equal in length. Now, let us study about the perpendicular distance of the chord from the centre as its length increases or decreases.

5 ircle-hord roperties 347 bserve the given figure, measure the length of the chords, their perpendicular distance from the centre and write the values in the table. Example Length of erpendicular distance the chord from the centre From the above data, we can observe that, E (a) as the length of the chord increases, the perpendicular distance... (b) as the length of the chord decreases, the perpendicular distance... X F (c) the perpendicular distance between the biggest chord and the centre is... Therefore, we can state that, If the length of the chord increases, its perpendicular distance from the centre decreases ; If the length of the chord decreases, its perpendicular distance from the centre increases. The perpendicular distance between the biggest chord and the centre is zero SEGMENT F ILE bserve the following figures and answer the given questions. 1. In figure 1 identify and name the minor arc and major arc. 2. What is called in figure 2? 3. In the given figures name the minor seg ment, major segment and semicircular region. ngles in a segment X 1. = = 2. EF = = 3. = = 4. X = Z bserve the following figure: Figure 1 In circle with centre, X is a chord, XZX is the minor segment and XX is the major segment. bserve the angles X and X. We say the angles are in segments. X is an angle in major segment XX. X is an angle in minor segment XZX. S Figure 2 Let us verify the properties of angles in a segment.

6 348 UNIT-14 ctivity 1 : bserve the angles in the minor segment. Measure the angles in the minor segment and write the measurements in the given table. Figure ngle in the minor Type of the segment angle From the above measurements we can state that, In a circle, angles in the minor segment are obtuse angles ctivity 2 : bserve the angles in major segments. Measure the angles and write the measurements in the table. N Figure ngle in the minor Type of the segment angle 1. = 2. LKN = From the above data, we can state that In a circle, angles in the major segment are acute angles ctivity 3 : bserve the following figures. Measure the angles and write the measurements in the table. Figure ngle in the minor Type of the segment angle 1. XZ = 2. S = From the above observations, we find that ngles in a semi circle are right angles ctivity 4 : Measure the angles and write the measurements in the table given below. bserve the three circles given below. T S (i) (ii) (iii) X E Fig.1 X G Fig.1 H Z L Fig.1 S I Fig.2 Fig.2 K Fig.2

7 ircle-hord roperties 349 Figure ngles in the same segment onclusion i T = T = ST = i i = = E = iii XG = XH = XI = From the above observations we can find that all angles in the same segment are equal. * all obtuse angles in a minor segment are equal. * all acute angles in a major segment are equal. * all angles in a semi-circle are equal to Hence, we can state that, NENTI ILES : In a circle, angles in the same segment are equal. bserve the figure and answer the following questions. 1. What are 1, 2, 3 and 4 in the figure? 2. What is '' with respect to the above figures? 3. Measure,, and. re they equal? We find that the circles have different radii but same centre. Such circles are called concentric circles. ircles having the same centre but different radii are called concentric circles. ongruent circles : bserve the figures given below. Measure the radii of the circles 1, 2 and 3. ou find that they are equal. ut the centres are different. E F ircles having same radii but different centres are called congruent circles.

8 350 UNIT-14 EXEISE Identify concentric circles and congruent circles in each of the following. 3cm cm 3cm 3cm 2cm cm 1.5cm 1cm 1cm 2cm 3 2. raw two concentric circles with centre '' and radii 2 cm and 3cm. 3. raw three concentric circles of radii 1.5 cm, 2.5cm and 3.5cm with as centre. 4. With and as centres draw two circles of radii 3.5 cm. 5. With 1 and 2 as centres draw two circles of same radii 3 cm and with the distance between the two centres equal to 5 cm. 6. raw a line segment = 8 cm and mark its mid point as. With 2 cm as radius draw three circles having, and as centres. With as centre and 4 cm radius draw another circle. Identify and name the concentric circles and congruent circles. 7. raw a circle of radius 4 cm and construct a chord of 6 cm length in it. raw two angles in major segment and two angles in minor segment. Verify that angles in major segment are acute angles and angles in minor segment are obtuse angles by measuring them. 8. raw a circle with centre and radius 3.5 cm and draw diameter in it. raw angles in semi-circles on either side of the diameter. Measure the angles and verify that they are right angles cm 3cm cm ircle - chord properties onstruction of chords of given length Verification of properties oncentric ircles ongruent ircles hord property Equal chords are equidistant from the centre ngles in a segment Major segment Minor segment onstruction acute angles obtuse angles Semi circle ight angles Same segment Equal