CHORD AND ANGLE PROPERTIES OF CIRCLES

[Pages:7]CHORD AND ANGLE PROPERTIES OF CIRCLES

ASSIGNMENT - 1

1. A chord of length 14 cm is at a distance of 24 cm from the centre of a circle. Calculate the radius of the circle. 2. In the figure, PQ is the diameter which meets the chord RS in T such that RT = TS = 4 cm. If QT = 3 cm, find the radius

of the circle.

3. A chord of length 16 cm is at a distance of 15 cm from the centre of the circle. Find the length of the chord of the same circle which is at a distance of 8 cm from the centre.

4. A chord of length 24 cm is at a distance of 5 cm from the centre of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the centre.

5. Two parallel chords of a circle of lengths 24 cm and 10 cm lie on the same side of its centre. Calculate the radius of the circle if the distance between the chords is 7 cm.

6. Two parallel chords of length 12 cm and 22 cm are on the same side of the centre of a circle of radius 61 cm. Find the distance between the chords.

7. In the above question, if the chords are on opposite sides of the centre of the circle, find the distance between them. 8. Two parallel chords of length 1.4 cm and 4 cm are 3.9 cm apart and lie on the opposite sides of the centre of the circle. Find

the radius of the circle. 9. In figure, O is the centre of the circles. PQ is a straight line and OT PQ. If the radius of the bigger circle be 34 cm, radius

of the smaller circle be 20 cm and OT = 16 cm, find the length of PR.

10. PQ is a diameter of a circle. X is a point in PQ such that PX = 8 cm and XQ = 18 cm. Find the length of the shortest chord through X.

11. In figure, O is the centre of the circle and diameter CD bisects the chord AB at E. If AE = EB = 8 cm and ED = 4 cm, find the radius of the circle.

12. An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle. 1

13. In figure, X and Y are the centres of the two inter-secting circles. If PX = 17 cm, PY = 25 cm and PQ = 30 cm, find the length of XY.

14. Chords AB and CD of a circle with centre O intersect at E. If OE bisects AED, prove that AB = CD. 15. Prove that the line through the centres of two intersecting circles bisects the common chord at right angles. 16. Prove that the line joining mid-points of two parallel chords of a circle passes through the centre of the circle. 17. In figure, AB is a chord of a circle with centre O and BC is a diameter. If OD AB, show that CA = 2OD and CA || OD.

18. Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it.

19. In the given figure, X and Y are mid-points of two equal chords PQ and RS of a circle with centre O. Prove that (i) OXY = OYX (ii) PXY = RYX

20. PQR is a right triangle with PR as the hypotenuse. Prove that the mid-point of PR is the centre of the circle that circumscribes PQR.

21. Prove that the opposite sides of a rectangle inscribed in a circle are equidistant from the centre. 22. Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre

of the circle. 23. If two equal chords of a circle intersect, then prove that their segments are equal. 24. Prove that the perpendicular bisector of a chord of a circle always passes through the centre. 25. In an equilateral triangle, prove that the centroid and the circumcentre of the triangle coincide. 26. A straight line is drawn cutting two equal circles and passing through the mid-point M of the line joining their centres O and

O. Prove that chords AB and CD, which are intercepted by two circles are equal.

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27. Two circles intersect at P and R. Two straight lines AB and CD are drawn through P and R respectively such that AB || CD. Prove that AB = CD.

28. ABC is an equilateral triangle. A circle is drawn with centre A so that it cuts AB and AC at M and N respectively. Prove that BN = CM.

29. Two circles having their centres at O and P intersect each other at Q and R. Two straight lines AQB and CRD are drawn parallel to OP. Prove that (i) AB = 2OP (ii) AB = CD.

1. 25 cm 6. 71 cm 11. 10 cm

1

2.

4 6

cm

7. 2.5 cm

12. 3 3 cm

ANSWERS

3. 30 cm 8. 17 cm 13. 28 cm

4. 10 cm 9. 18 cm

5. 49 cm 10. 24 cm

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1. In the given figure, O is the centre of the circle, BAD = 75? and chord BC = chord CD. Find (i) BOC (ii) OBD (iii) BCD.

2. In the figure, AB is parallel to DC, BCE = 80? and BAC = 25?. Find :

ASSIGNMENT - 2

7. A circle with centre O, diameter AB and a chord AD is drawn. Another circle is drawn with AO as diameter to cut AD at C. Prove that BD = 2 OC.

8. In the figure, O is the centre of the circle and AOC = 160?. Prove that 3 y ? 2 x = 140?.

9. From the figure, find (i) APB (ii) AOB.

10. In the given circle with diameter AB, find the value of x.

3. In the figure, PQ = QR, RQP = 68?, PC and CQ

are tangents to the circle with centre O. Calculate the values of :

(ii) QCP

4. In the given figure, O is the centre of the circle and PBA = 45?. Calculate the value of PQB.

5. In the figure, if ACE = 43? and CAF = 62?, find the values of a, b and c.

A

E

F

6. In the figure, BAD = 65?, ABD = 70? and BDC = 45?. Find :

(i) BCD

(ii) ADB

Hence, show that AC is a diameter.

11. In the figure, AB is the diameter of a circle with centre O. BCD = 120?. Find : (i) DBA (ii) BAD.

12. In the figure, AE is the diameter of a circle. Write down the numerical value of ABC + CDE. Give reasons for your answer.

13. In the figure, AD is the diameter of the circle. If BCD = 130?, calculate : (i) DAB (ii) ADB

14. In the figure, A, D, B, C are four points on the circumference of a circle with centre O. Arc AB = 2 arc BC and AOB = 108?. Calculate in degrees (i) ACB (ii) CAB (iii) ADB

15. In figure, AB = AC = CD, ADC = 38?. Calculate

(i) ABC (ii) BEC

ANSWERS

1. (i) 75? (ii) 15? (iii) 105? 2. (i) 55? (ii) 55? (iii) 100? 3. (i) 112? (ii) 68? 4. 45? 5. a = 105?, b = 13? and c = 62? 6. (i) 115? (ii) 45? 9. (i) 80? (ii) 120? 10. 60? 11. (i) 30? (ii) 60? 12. 270? 13. (i) 50? (ii) 40? 14. (i) 54? (ii) 27? (iii) 126?

15. (i) 76? (ii) 28? 4

ASSIGNMENT - 3

1. In each of the following figures, if O is the centre of the

6. In the figure, O is the centre of the circle. If AOB =

circle, find the values of x and y.

140? and OAC = 50?, find (i) ABC (ii) BCO

(iii) OAB (iv) BCA.

7. In the figure, AC is the diameter of the circle. If DAB = 120? and ACB = 40?, calculate ADB and ABD.

2. In the figure, O is the centre of the circle and AB is a diameter. If AB || DC and BAC = 20?, find the values of p, q, r and s.

8. In the figure, two circles intersect at A and B. The centre of one circle lies on the circumference of the other. If AOB = 84?, find the value of x.

3. In the figure, find ABD.

9. In the figure, ABCD is a cyclic quadrilateral. If BCF = 48? and x = 2y, find the numerical values of x and y.

4. In the figure, ABCD is a cyclic quadrilateral and DF || BC. If EDF = 15? and BAD = 80?, find ABC.

5. In the figure, find the values of a, b, c and d.

10. In the figure, AB is the diameter of the circle. If APB = 120? and CBP = 25?, find ADB.

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11. In the figure, AB and CD pass through the centre O of the circle. If EBO = 40? and AOC = 75?, find the values of x and y.

12. In the figure, O is the centre of the circle and a pentagon ABCDE is inscribed in it. If AB = BC = CD and ABC = 130?, find (i) AEB (ii) AED (iii) COD

17. PQRS is a cyclic quadrilateral in which QPR = RPS and QRP = SRP. Show that PR is a diameter of the circle.

18. Two circles with centres O and O intersect at points A and B. AC and AD are their diameters respectively. Prove that the points C, B and D are collinear.

19. PQ is a chord of a circle with centre O. R is any point on the minor arc PQ. Prove that POQ + 2 PRQ = 360?.

20. Prove that quadrilateral formed by angle bisectors of a cyclic quadrilateral ABCD is also cyclic.

21. AB and BC are two equal chords of a circle. Prove that the bisector of ABC passes through the centre.

22. In the figure, AB and CD are two chords of the circle, with centre O which intersect at right angles. Show that OAD = BDC.

13. In the figure, P and Q are the centres of two circles intersecting at B and C. ACD is a straight line. Calculate the numerical value of x.

14. In the figure, O is the centre of the circle. If AOE = 150? and DAO = 51?, calculate the magnitude of (i) CEB (ii) CBE.

23. Prove that the rhombus inscribed in a circle is a square.

24. ABCD is a cyclic quadrilateral in which AB and DC on being produced, meet at P such that PA = PD. Prove that AD is parallel to BC.

25. Prove that an isosceles trapezium is always cyclic.

26. C is a point on the minor arc AB of a circle with centre O. Given ACB = x?, AOB = y?, express y in terms of x. Calculate x if ACBO is a parallelogram.

15. In the figure, length of arc AB and arc BC are in the ratio 3 : 2. If AOB = 96?, find (i) BAC (ii) ADB.

27. PQRS is a cyclic quadrilateral in which PQ = RS. Prove that PQR = SRQ.

28. X is a point on a circle with centre O. If X is equidistant from the two radii OP, OQ, then prove that arc PX = arc QX.

29. In the figure, ABC is an isosceles triangle inscribed in a circle with centre O. If AB = AC, prove that AP bisects BPC.

16. ABCD is a quadrilateral in which the circle through A, B and C has D as its centre. AB is produced to any point E. If ADC = 136?, find CBE.

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30. If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.

31. In the figure, ABC is an isosceles triangle with AB = AC. A circle is drawn on AB as diameter. Prove that the circle bisects BC.

a circle with centre O. Find (i) AOB (ii) ACB (iii) ABC.

32. In the figure, AB is a side of a regular 6-sided polygon and AC is a side of a regular 8-sided polygon inscribed in

33. Prove that the points A, B, C and D are concyclic, if the line segments AB and CD intersect at a point P such that AP PB = CP PD.

ANSWERS

1. (i) 75? (ii) 128? (iii) 48? (iv) 115?, 65? 2. 40?, 100?, 50?, 110? 3. 110? 4. 115? 5. 34?, 56?, 68?, 56? 6. (i) 40? (ii) 60? (iii) 20? (iv) 110? 7. 40?, 20? 8. 69? 9. 50?, 28? 10. 95? 11. 50?, 25? 12. (i) 25? (ii) 75? (iii) 50? 13. 130? 14. (i) 51? (ii) 105? 15. (i) 32? (ii) 132? 16. 68? 26. y = 2 (180? ? x), x = 120? 32. (i) 60? (ii) 30? (iii) 22.5?

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