10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)


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1 10. Circles Q 1 True or False: It is possible to draw two circles passing through three given noncollinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular bisector of two chords of a circle intersect at centre of the circle. Mark (1) Q 3 True or False : If two arcs of a circle are congruent, then corresponding chords are unequal. Mark (1) Q 4 State the following statement as true or false. Give reasons also. Line segment joining the centre to any point on the circle is a radius of the circle. Mark (1) Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Q 6 In figure, O is the centre of a circle and ADC= Find BAC. Q 7 In Figure, ABC is equilateral. Find (i) ABC (ii) AEC. IX Mathematics C.B.S.E. Practice Papers Page 70
2 Q 8 The diameter of a circle is 5 cm. If a chord is 4 cm long then find the distance between the centre and the chord of the circle. Q 9 The chord of a unit (in cm) circle subtends an angle of 120 at the centre. Find length of the chord in cm. Q 10 Find the value of CAB in the figure given below. Q 11 If ON and OM are perpendiculars to CD and AB respectively,then find the length of AM in the figure given below. Q 12 Find the value of x in the figure given below, where BC = 6 cm and AB = 10 cm. Q 13 Prove that diameter is the greatest chord in a circle. IX Mathematics C.B.S.E. Practice Papers Page 71
3 Q 14 ABCD is a cyclic quadrilateral in which AB CD. If B=65 0, then find other angles. Q 15 Find the value of a and b in the figure given below where SOT=30. Q 16 Find the value of angle CAB in the figure given below. Q 17 Find the value of x in the figure given below, where BC = 6 cm and AB = 10 cm. Q 18 Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB. Q 19 O is the centre of a circle and the measure of arc ABC is 100. Determine ADC and ABC. IX Mathematics C.B.S.E. Practice Papers Page 72
4 Q 20 If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD. Q 21 Prove that the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. Q 22 Prove that equal chords of a circle subtend equal angles at the centre. Q 23 Bisector AD of BAC of ABC passes through the center O of the circumcircle of ABC. Prove that AB = AC. Q 24 A, B, C and D are the four points on a circle. AC and BD intersect at point E such that BEC = 130 and ECD = 20. Find BAC. Q 25 A,B,C and D are four consecutive points on a circle such that AB = CD. Prove that AC = BD. IX Mathematics C.B.S.E. Practice Papers Page 73
5 Q 26 Find the value of a 0 and b 0 in the figure given below. Q 27 If a chord of length 24 cm is at a distance of 5 cm from the centre in the circle, then find the area of the circle. Q 28 Prove that all the chords of a circle through a given point within it, the least is one which is bisected at that point. Q 29 AB and CD are the two chords of the circle such that AB = 6 cm, CD = 12 cm and AB CD. If the distance between AB and CD is 3 cm, find the radius of the circle. Marks (4) Q 30 A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each telephone. Marks (4) Q 31 Two equal chords AB and CD of circle with center O, when produced meet at a point E. Prove that BE = DE and AE = CE. Marks (4) Q 32 In a circle with centre O, chords AB and CD intersect inside the circle at E. Prove that AOC + BOD = 2 AEC. Marks (4) Q 33 OA and OB are respectively perpendicular to chords CD and EF of a circle whose centre is O. If OA = OB, prove that CD = EF. Marks (4) IX Mathematics C.B.S.E. Practice Papers Page 74
6 Most Important Questions Q 1 What is a Circle? Q 2 What is the fixed point and fixed distance in a circle are called. Q 3 In how many parts a circle divides the plane on which it lies. Name them. Q 4 What is diameter and what is its relation with the radius. Q 5 Define Arc, Major arc and Minor arc, Semicircle. (a) The centre of a circle lies in the interior of the circle. (b) The longest chord of a circle is the diameter of circle. (c) Segment of a circle is the region between an arc and two radii. (d) Sector of a circle is the region between a chord and either of its arc. (e) Line from the centre to any point on the circle is radius. Q 7 If major arc and minor arc are equal. What we call these arcs, and the regions formed. Q 8 There are two equal chords of length 5 cm each in a circle. One chord subtends an angle of 50 0 at the centre. What is the angle subtended by other chord at the centre. Q 9 Two chords make angles of 55 0 at the centre, if one chord is 7 cm long what is the length of another chord. Q 10 In the given figure, OL perpendicular to AB, if AL = 3 cm, find BL. IX Mathematics C.B.S.E. Practice Papers Page 75
7 Q 11 In the given figure below AL = 5cm and BL = 5cm. Find OLA. Q 12 In a circle of radius 5cm how far from the centre will be a chord of length 6cm. Q 13 How many circles can be drawn through (a) One given point (b) Two given points (c) Three given points Q 14 If a circle is given, how can you find its centre using suitable construction. Q 15 An arc of a circle is given, complete the circle. Q 16 In the figure given below AB and CD are chords and PQ is the diameter.if AEQ = DEQ, prove that AB = CD. Q 17 If two circles intersect at two points, prove that the perpendicular bisector of the common chord will pass through their centres. Q 18 If two circles intersect at two points, prove that the line through their centres is the perpendicular bisector of the common chord. IX Mathematics C.B.S.E. Practice Papers Page 76
8 Q 19 Two circles having radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord. Q 20 Three boys are sitting on the circumference of a circular park with equal distance between them. If radius of the park is 20m find the linear distance between them. IX Mathematics C.B.S.E. Practice Papers Page 77
9 Q 21 A chord meets two concentric circles at points A, B, C & D as shown in the figure below. Prove that AC = BD. Q 22 Two equal chords intersect within the circle, prove that the line joining the point of intersection and the centre makes equal angles with the chords. Q 23 In the given figure below, DBC = 45 and BAC = 45. Find BCD. Q 24 In the figure shown, AD and AC are the diameters of the circles. Prove that the intersection point of circles lie on the third side of the triangle ACD. IX Mathematics C.B.S.E. Practice Papers Page 78
10 Q 25 Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic. Q 26 In the given figure, find BDC. Q 27 A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. Q 28 In the given figure, find BAC. IX Mathematics C.B.S.E. Practice Papers Page 79
11 Q 29 In the figure below find BCD. Further if AB = BC find ECD. Q 30 If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. Q 31 If nonparallel sides of a trapezium are equal, prove that it is cyclic. Q 32 ABC and ADC are two right triangles with common hypotenuse AC. Prove that CAD = CBD. IX Mathematics C.B.S.E. Practice Papers Page 80
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