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.
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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.
ANSWERS
6. 71 cm
1
cm
6
7. 2.5 cm
11. 10 cm
12. 3 3 cm
1. 25 cm
2. 4
3. 30 cm
4. 10 cm
5. 49 cm
8. 17 cm
9. 18 cm
10. 24 cm
13. 28 cm
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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°.
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 :






9. From the figure,
find (i) APB
(ii) AOB.
10. In the given circle with diameter AB,
find the value of x.
11. In the figure, AB is the
diameter of a circle with
centre O. BCD = 120°.
Find : (i) DBA
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
(ii)
12. In the figure, AE is the diameter of a circle. Write
down the numerical value
of ABC + CDE. Give
reasons for your answer.
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
the values of a, b and c.
A
13. In the figure, AD is the diameter of the circle. If BCD =
130°, calculate : (i) DAB
(ii) ADB
CAF = 62°, find
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
F
E
6. In the figure, BAD = 65°,
and BDC = 45°. Find :
BAD.
ABD = 70°
(i) BCD
(ii) ADB
Hence, show that AC is a diameter.
(ii)
BEC
ANSWERS
1.
(i) 75° (ii) 15° (iii) 105° 2. (i) 55° (ii) 55° (iii) 100° 3. (i) 112° (ii) 68°
6. (i) 115° (ii) 45°
4. 45°
5. a = 105°, b = 13° and c = 62°
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°
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ASSIGNMENT - 3
1. In each of the following figures, if O is the centre of the
circle, find the values of x and y.
6. In the figure, O is the centre of the circle. If AOB =
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.
3. In the figure, find
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.
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.
10. In the figure, AB is the diameter of the circle. If
= 120° and CBP = 25°, find ADB.
5. In the figure, find the values of a, b, c and d.
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APB
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.
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.
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
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.
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.
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.
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.
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.
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.
a circle with centre O. Find (i)
(iii) ABC.
AOB
(ii)
ACB
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.
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.
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
ANSWERS
1. (i) 75° (ii) 128° (iii) 48° (iv) 115°, 65° 2. 40°, 100°, 50°, 110°
6. (i) 40° (ii) 60° (iii) 20° (iv) 110°
12. (i) 25° (ii) 75° (iii) 50° 13. 130°
7. 40°, 20°
8. 69°
3. 110°
9. 50°, 28°
4. 115°
10. 95°
5. 34°, 56°, 68°, 56°
11. 50°, 25°
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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