sa2 ix maths(ch 10)

Transcription

sa2 ix maths(ch 10)

Assignments in Mathematics Class IX (Term 2)
10. CIRCLES
IMPORTANT TERMS, DEFINITIONS AND RESULTS
The collection of all the points in a plane, which
are at a fixed distance from a fixed point in the plane,
is called a circle.
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The fixed point is called the centre of the circle
and the fixed distance is called the radius of the
circle.
In the given figure, O is the centre and the length
OP is the radius of the circle.
A circle divides the plane on which it lies into three
parts. They are : (i) inside the circle, which is also
called the interior of the circle; (ii) the circle and
(iii) outside the circle, which is also called the
exterior of the circle. The circle and its interior
make up the circular region.
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points P and Q in the given figure. You find that
there are two pieces, one longer and the other
smaller. The longer one is called the major arc PQ
and the shorter one is called the minor arc PQ.
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The length of the complete circle is called its
circumference. The region between a chord and
either of its arcs is called a segment of the circular
region or simply a segment of the circle. You will
find that there are two types of segments also, which
are the major segment and the minor segment.
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The region between an arc and the two radii, joining
the centre to the end points of the arc is called a
sector. Like segments, you find that the minor arc
corresponds to the minor sector and the major arc
corresponds to the major sector.
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Equal chords of a circle subtend equal angles at the
centre.
If the angles subtended by the chords of a circle at
the centre are equal, then the chords are equal.
The perpendicular from the centre of a circle to a
chord bisects the chord.
The line drawn through the centre of a circle to bisect
a chord is perpendicular to the chord.
There is one and only one circle passing through
three given non-collinear points.
A chord of a circle is a line segment joining any
two points on the circle. In the given figure PQ, RS
and AOB are the chords of a circle.
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A diameter is a chord of a circle passing through
the centre of the circle. In the given figure, AOB is
the diameter of the circle. A diameter is the longest
chord of a circle.
Diameter = 2 × radius
A piece of a circle between two points is called an
arc. Look at the pieces of the circle between two
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Equal chords of a circle (or of congruent circles)
are equidistant from the centre (or centres).
Chords equidistant from the centre of a circle are
equal in length.
The angle subtended by an arc at the centre is double
the angle subtended by it at any point on the
remaining part of the circle.
Angles in the same segment of a circle are equal.
Points which lie on the same circle are called
concyclic points.
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A quadrilateral is said to be a cyclic quadrilateral if
there is a circle passing through all its four vertices.
If a line segment joining two points subtends equal
angles at two other points lying on the same side of
the line containing the line segment, the four points
lie on a circle (i.e., they are concyclic).
The sum of either pair of opposite angles of a cyclic
quadrilateral is 180°.
If the sum of a pair of opposite angles of a
quadrilateral is 180°, the quadrilateral is cyclic.
SUMMATIVE ASSESSMENT
MULTIPLE CHOICE QUESTIONS
[1 Mark]
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A. Important Questions
(c) interior opposite angle
(d) corresponding angle
5. Three chords AB, CD and EF of a circle are
respectively 3 cm, 3.5 cm and 3.8 cm away from
the centre. Then which of the following relations
is correct ?
(a) AB > CD > EF
(b) AB < CD < EF
(c) AB = CD = EF
(d) none of these
1. In the figure, O is the centre of the circle with AB
as diameter. If ∠AOC = 40°, the value of x is
equal to :
(b) 60°
(c) 70°
(d) 80°
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(a) 50°
6. In the given figure, O is the centre of the circle.
If ∠CAB = 40° and ∠CBA = 110°, the value of
x is :
2. Read the following two statements and choose the
correct option.
Statement I : Diameter is the longest chord of a
circle.
Statement II : A circle has only finite number of
equal chords.
(a) only I is true
(b) only II is true
(c) both I and II are true
(d) neither I nor II is true
3. In the given figure, O is the centre of the circle.
If OA = 5 cm and OC = 3 cm, then the length of
AB is :
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(a) 50°
(b) 80°
(c) 55°
(d) 60°
7. In a circle, chord AB of length 6 cm is at a distance of
4 cm from the centre O. The length of another chord
CD which is also 4 cm away from the centre is :
(a) 6 cm
(b) 4 cm
(c) 8 cm (d) 3 cm
8. In the figure, chord AB is greater than chord CD.
OL and OM are the perpendiculars from the centre
O on these two chords as shown in the figure. The
correct releation between OL and OM is :
(a) 4 cm
(b) 6 cm
(c) 8 cm (d) 15 cm
4. If one side of a cyclic quadrilateral is produced,
then the exterior angle is equal to its :
(a) exterior adjacent angle
(b) alternate angle
(a) OL = OM
(c) OL > OM
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(b) OL < OM
(d) none of these
9. Number of circles passing through two given
points is :
(a) one
(b) two
(c) finite (d) infinite
10. The length of a chord in a circle of diameter
10 cm is 6 cm. The distance of the chord from its
centre is :
(a) 5 cm
(b) 3 cm
(c) 8 cm (d) 4 cm
16. In the figure, if AOB is a diameter of the circle
and AC = BC, then ∠CAB is equal to :
11. In the given figure, ABC is
an equilateral triangle. Then
measure of ∠BEC is :
(a) 100°
(b) 120°
(c) 140°
(d) 90°
(a) 30°
(b) 60°
(c) 90°
(d) 45°
17. AD is a diameter of a circle and AB is a chord.
If AD = 34 cm, AB = 30 cm, then the distance of
AB from the centre of the circle is :
(a) 17 cm (b) 15 cm (c) 4 cm (d) 8 cm
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12. Two chords AB and CD subtend x° each at the
centre of the circle. If chord AB = 8 cm, then
chord CD is :
18. If AB = 12 cm, BC = 16 cm and AB is
perpendicular to BC, then the radius of the circle
passing through the points A, B and C is :
(a) 6 cm
(b) 8 cm
(c) 10 cm (d) 12 cm
19. In the figure, O is the centre of the circle. If
∠OAB = 40°, then ∠ACB is equal to :
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(a) 4 cm
(b) 8 cm
(c) 16 cm (d) 12 cm
13. The radius of a circle is 10 cm and the length of
the chord is 12 cm.The distance of the chord from
the centre is :
(a) 12 cm (b) 10 cm (c) 8 cm (d) 13 cm
14. In the figure, O is the centre of the circle and
∠AOB = 80°. The value of x is :
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(a) 50°
(b) 40°
(c) 60°
(d) 70°
20. In the figure , if ∠DAB = 60°, ∠ABD = 50°, then
∠ACB is equal to :
(a) 30°
(b) 40°
(c) 60°
(d) 160°
15. In the given figure, a circle with centre O is shown,
where ON > OM. Then which of the following
relations is true between the chord AB and chord
CD ?
(a) AB = CD
(c) AB < CD
(a) 60°
(b) 50°
(c) 70°
(d) 80°
21. In the figure, O is the centre of the circle. If ∠ABC
= 20°, then ∠AOC is equal to :
(b) AB > CD
(d) none of these
(a) 20°
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(b) 40°
(c) 60°
(d) 10°
22. In the figure, if ∠SPR = 73°, ∠SRP = 42°, then
∠PQR is equal to :
(a) 65°
(b) 70°
(c) 74°
∠PQR = 100°. Then the reflex ∠POR is :
(d) 76°
23. In the figure, O is the centre of the circle. If ∠OPQ
= 25° and ∠ORQ = 20°, then the measures of
∠POR and ∠PQR respectively are :
(b) 105°, 450°
(d) 100°, 50°
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(a) 90°, 45°
(c) 110°, 55°
(a) 280°
(b) 200° (c) 260° (d) none of these
27. In the given figure, E is any point in the interior
of the circle with centre O. Chord AB =
Chord AC. If ∠OBE = 20°, then the value of x
is :
(a) 40°
(b) 45°
(c) 50°
(d) 70°
∠AOB = 60°. The value of x is :
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24. Two circles intersect at the points A and B. AD
and AC are diameters of the respsective circles as
shown in the following figure. Sum of ∠ABD and
∠ABC :
(a) 30°
(b) 35°
(c) 25°
(d) 40°
29. In the figure, AB and CD are two chords of a
circle with centre O and MN as diameter. They
intersect at a point E. If ∠AEN = ∠DEN = 45°
and AB = 6.5 cm, then the length of chord CD is
equal to :
(a) is greater than 180° (b) is equal to 180°
(c) is less than 180° (d) has no definite value
25. In the figure, if ∠CAB = 40° and AC = BC, then
∠ADB equal to :
(a) 40°
(b) 60°
(c) 80°
(a) 13 cm
(c) 7.0 cm
(d) 100°
4
(b) 6.5 cm
(d) none of these
30. In the figure, points A, B, C and D lie on a circle.
BC is produced to P and ∠BAD = 100°. The
measure of ∠DCP is :
35. In the figure, O is the centre of the circle of radius
5 cm. OP ⊥ AB, OQ ⊥ CD, AB || CD, AB = 8 cm
and CD = 6 cm. The length of PQ is :
(a) 100°
(b) 180°
(c) 110° (d) 90°
31. In the figure, chord DE is parallel to the diameter
AC of the circle. If ∠CBE = 60°, then the measure
of ∠CED is :
(a) 8 cm
(c) 6 cm
(b) 1 cm
(d) none of these
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∠ABD = 45°. The value of x is :
(a) 90°
(b) 60°
(c) 30°
(d) 50°
∠ABP = 40°. The measure of ∠PQB is :
(b) 45°
(d) none of these
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(a) 90°
(c) 135°
∠AOC = 130°. Then ∠ADC is :
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(a) 40°
(b) 50°
(c) 100° (d) 80°
33. In the figure, A, B, C and D are four points on a
circle. AC and BD intersect at a point E. If
∠BEC = 140° and ∠ECD = 30°, then the value of
∠BAC is :
(a) 65°
(c) 130°
(b) 230°
(d) 115°
38. In the figure, ∠AOB = 90° and ∠ABC = 30°,
then ∠CAO is equal to :
(a) 110°
(b) 120°
(c) 100° (d) 90°
∠AOC = 130°. The value of x is :
(a) 25°
(b) 50°
(c) 40°
(d) 35°
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(a) 30°
(b) 45°
(c) 90°
(d) 60°
39. In the figure, AB and CD are two equal chords of
a circle with centre O, OP and OQ are
perpendiculars on chords AB and CD, respectively.
If ∠POQ = 150°, then ∠APQ is equal to :
(a) 30°
(b) 75°
(c) 15°
(a) 30°
(b) 45°
(c) 60°
(d) 120°
42. In the figure, two congruent circles have centres
O and O′. Arc AXB subtends an angle of 75° at
the centre O and arc A′ Y B′ subtends an angle of
25° at the centre O′. Then the ratio of arcs AXB
and A′ Y B′ is :
(d) 60°
(a) 80°
(b) 50°
(c) 40°
(d) 30°
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40. ABCD is a cyclic quadrilateral such that AB is a
diameter of the circle circumscribing it and ∠ADC
= 140°, then ∠BAC is equal to :
41. In the figure, BC is a diameter of the circle and
∠BAO = 60°. Then ∠ADC is equal to :
(a) 2 : 1
(b) 1 : 2
(c) 3 : 1
(d) 1 : 3
B. Questions From CBSE Examination Papers
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1. In the figure, if ∠ACB = 50°, then ∠OAB is :
[T-II (2011)]
(a) 225°
(b) 128°
(c) 150° (d) 75°
4. Which of the following pairs of angles are opposite
angles of a cyclic quadrilateral?
[T-II (2011)]
(a) 131°, 28°
(b) 95°, 55°
(c) 123°, 57°
(d) 64°, 52°
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(a) 50°
(b) 40°
(c) 60°
(d) 70°
2. In the figure, if OA = 17 cm, AB = 30 cm and OD
is perpendicular to AB, then CD is equal to :
[T-II (2011)]
(a) 8 cm
(b) 9 cm
5. A circle divides a plane in which it lies including
itself in :
[T-II (2011)]
(a) 2 parts (b) 3 parts (c) 4 parts (d) 5 parts
6. O is the centre of the circle ∠QPS = 65°; ∠PRS
= 33°′, ∠PSQ is :
[T-II (2011)]
(c) 10 cm (d) 11 cm
3. ABCE is a cyclic quadrilateral. O is the centre of
the circle and ∠AOC = 150°, then ∠CBD :
[T-II (2011)]
(a) 90°
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(b) 82°
(c) 102°
(d) 42°
7. In the figure, O is the centre of the circle. ∠POQ
= 100°; ∠POR = 110°; then ∠QPR equals :
[T-II (2011)]
12. In the figure, O is the centre of the circle, find
∠AOC, given ∠BAO = 30° and ∠BCO = 40°.
[T-II (2011)]
(a) 210°
(b) 200°
(c) 150° (d) 75°
8. In the figure ∠CAB = 45° ; ∠DBC = 55°, then
∠DCB equals :
[T-II (2011)]
(a) 35°
(c) 70°
(b) 140°
(d) cannot be determined
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13. In the figure, O is the centre of a circle and ∠OBA
= 60°. Then ∠ACB equals :
[T-II (2011)]
(a) 55°
(b) 80°
(c) 100° (d) 120°
9. In the figure, PQ is the diameter of the semicircle
in which ∠SPR = 30° ; ∠QPR = 20° ; then ∠SRP
equals :
[T-II (2011)]
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(a) 60°
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(a) 30°
(a) 26 cm (b) 10 cm (c) 5 cm (d) 8 cm
11. In the figure, if O is the centre of the circle;
OL = 4 cm, AB = 6 cm and OM = 3 cm, then
CD is equal to :
[T-II (2011)]
(b) 8 cm
(c) 6 cm
(c) 75°
(d) 30°
14. In the figure, AOB is a diameter of the circle and
AC = BC. Then ∠CAB is :
[T-II (2011)]
(a) 40°
(b) 65°
(c) 120° (d) 35°
10. In the figure, O is the centre of the circle of radius
13 cm and chord AB is of length 24 cm. If OC is
perpendicular from the centre to AB, then OC
equals :
[T-II (2011)]
(a) 4 cm
(b) 120°
(b) 45°
(c) 60°
(d) 90°
15. If AOB is the diameter of the circle and
∠B = 35°, then x equals :
[T-II (2011)]
(a) 90°
(d) 10 cm
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(b) 55°
(c) 75°
(d) 45°
23. In the given figure, if POQ is a diameter of the
circle and PR = QR, then ∠RPQ is equal to :
[T-II (2011)]
16. Given two concentric circles with centre O. A line
cuts the circles at A, B, C, D, respectively. If AB
= 10 cm, then length CD is :
[T-II (2011)]
(a) 5 cm
(b) 10 cm
(c) 7.5 cm
(d) none of these
17. In the figure, O is the centre of the circle, ∠CBE
= 25° and ∠DEA = 60°. The measure of ∠ADB
is :
[T-II (2011)]
(a) 90°
(b) 95°
(c) 85°
(d) 120°
18. Given three collinear points, then the number of
circles which can be drawn through these points
is :
[T-II (2011)]
(a) zero
(b) one
(c) two
(d) infinite
(a) 9°
(b) 10°
(c) 11°
(d) 12°
25. Two circles are said to be concentric, if :
[T-II (2011)]
(a) they have same radius
(b) they have different radii
(c) they have same centre
(d) their centres are collinear
26. In the figure, if O is the centre, then the value of
y is :
[T-II (2011)]
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19. The length of chord which is at a distance of
12 cm from centre of circle of radius 13 cm is :
[T-II (2011)]
(a) 5 cm
(b) 12 cm (c) 13 cm (d) 10 cm
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(a) 30°
(b) 60°
(c) 90°
(d) 45°
24. For what value of x in the figure, points A, B, C
and D are concyclic?
[T-II (2011)]
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20. In the figure, AB is a diameter of the circle.
CD || AB and ∠BAD = 40°, then ∠ACD is :
[T-II (2011)]
(a) 35°
(b) 75° + x (c) 70° – x (d) 140°
33. In a circle with centre O, chords AB and CD are
of length 5 cm and 6 cm respectively and subtend
angle x° and y° at centre of circle respectively
then :
[T-II (2011)]
(a) 40°
(b) 90°
(c) 130° (d) 140°
21. In the figure, the values of x and y respectively
are :
[T-II (2011)]
(a) 20°, 30° (b) 36°, 60° (c) 15°, 30° (d) 25°, 30°
22. The distance of a chord 8 cm long from the centre
of a circle of radius 5 cm is :
[T-II (2011)]
(a) 4 cm
(b) 3 cm
(c) 2 cm (d) 9 cm
(a) x° = y°
(c) x° > y°
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(b) x° < y°
(d) none of the above
28. ABCD is a cyclic quadrilateral as shown in the
figure. The value of (x + y) is : [T-II (2011)]
(a) 60°
(b) 80°
(c) 90°
(d) 40°
∠ABC = 36°. The measure of ∠AOC is :
[T-II (2011)]
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(a) 200°
(b) 100°
(c) 180° (d) 160°
29. AD is a diameter of a circle and AB is a chord.
If AD = 34 cm and AB = 30 cm, the distance of
AB from the centre of the circle is : [T-II (2011)]
(a) 17 cm (b) 15 cm (c) 4 cm (d) 8 cm
30. In the figure, if O is the centre and ∠BOA = 120°,
then the value of x is :
[T-II (2011)]
(a) 36°
(b) 72°
(c) 144
(d) 18°
37. In the figure, if AB is the diameter of the circle,
then the value of x is :
[T-II (2011)]
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(a) 120°
(b) 60°
(c) 30°
(d) 90°
31. If AB = 12 cm, BC = 16 cm and AB ⊥ BC, then
radius of the circle passing through A, B and C
is :
[T-II (2011)]
(a) 6 cm
(b) 8 cm
(c) 10 cm (d) 12 cm
32. An equilateral ∆ABC is inscribed in a circle with
centre O. The measure of ∠BOC is : [T-II (2011)]
(a) 110°
(b) 100°
(c) 120° (d) 130°
33. In the figure, quadrilateral PQRS is cyclic. If ∠P
= 80°, then ∠R is :
[T-II (2011)]
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(a) 40°
(b) 50°
(c) 80°
(d) 90°
38. In the figure, if ∠OAB = 40°, then ∠ACB is equal
to :
[T-II (2011)]
(a) 80°
(b) 40°
(c) 100° (d) 120°
∠ABC = 55°, then ∠ADC is :
[T-II (2011)]
(a) 50°
(b) 40°
(c) 60°
(d) 70°
39. In the figure, O is the centre of the circle with
∠AOB = 85° and ∠AOC = 115°, then ∠BAC
is :
[T-II (2011)]
(a) 55°
(b) 110°
(c) 75°
(d) 27.5°
35. In the figure, if ∠PQR = 40°, then the value of
∠PSR is :
[T-II (2011)]
(a) 115°
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(b) 85°
(c) 80°
(d) 100°
SHORT ANSWER TYPE QUESTIONS
[2 Marks]
1. Find the the length of a chord which is at a distance of 5 cm from the centre of a circle whose
radius is 13 cm.
2. The radius of a circle is 10 cm and a chord of the
circle is 12 cm long. Find the distance of the chord
from the centre of the circle.
3. Can we have a cyclic quadrialateral ABCD
such that ∠A = 90°, ∠B = 70°, ∠C = 95° and
∠D = 105° ?
4. Two congruent circles with centres O and O′
intersect at two points A and B. Check whether
∠AOB = ∠AO′ B or not.
5. In the figure, PQR is right angled at Q. Point S is
taken on side PR such that PS = SR and
QR = QS. Find the measure of ∠QSR.
9. Check whether the following statement is true.
A, B, C, D are four points such that
∠BAC = 45° and ∠BDC = 45°, then A, B, C, D
are concyclic.
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10. In the figure, if AOB is a diameter and
∠ADC = 120°, find ∠CAB.
11. In the figure, O is the centre of the circle, BD =
DC and ∠DBC = 30°. Find the measure of
∠BAC.
12. If arcs AXB and CYD of a circle are congruent,
find the ratio of AB and CD.
In the figure, ∆ABC is equilateral. Find ∠BDC
and ∠BEC.
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8.
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6. Show that diameter of a circle is the greatest chord.
7. In the figure, O is the centre of the circle. If ∠BAC
= 50°, find ∠BOC.
13. AOB is a diameter of a circle and C is a point on
the circle. Check whether AC2 + BC2 = AB2 is
true or not.
14. Two chords of a circle of lengths 10 cm and 8 cm
are at the distances 8 cm and 3.5 cm respectively
from the centre. Check whether the above
statement is true or not.
2. In the figure, ∠PQR
= 100°, where P, Q, R
are points on a circle,
with centre O. Find
∠OPR. [T-II (2011)]
1. In the figure, A, B, C,
D are four points on a
circle. AC and BD intersect at a point E such
that ∠BEC = 130° and
∠ECD = 20°. Find
∠BAC. [T-II (2011)]
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3. In the figure, ∠ABC = 69°; ∠ACB = 31°. Find
∠BDC.
[T-II (2011)]
9. In the figure, OD ⊥ AB and AC is a diameter.
Show BC = 2 (OD).
[T-II (2011)]
4. Prove that equal chords of a circle subtend equal
angles at the centre.
[T-II (2011)]
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5. AB and CD are two parallel chords on the same
side of the circle such that AB = 6 cm;
CD = 8 cm. The small chord is at a distance of 4
cm from the centre. At what distance from the
centre is the other chord.
[T-II (2011)]
10. Two concentric circles with centre O have A, B,
C and D as points of intersection with a line l as
shown in the figure, If AD = 12 cm and
BC = 8 cm, find the length of AB and CD.
[T-II (2011)]
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6. In ΔABE, AE = BE. Circle through A and B
intersects AE and BE at D and C. Prove that
DC | | AB.
[T-II (2011)]
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∠D = 130°, then find ∠CAB.
[T-II (2011)]
∠BOC = 120°, find ∠CDE.
[T-II (2011)]
12. In the figure, ∠PQR = 100°, where P, Q and R are
points on a circle with centre O. Find ∠OPR.
[T-II (2011)]
8. In the figure, AB is the diameter of the circle with
centre O. If ∠DAB = 70° and ∠DBC = 30°,
determine ∠ABD, ∠CDB.
[T-II (2011)]
11
13. ABCD is a cyclic quadrilateral. Find ∠ADB.
[T-II (2011)]
18. Suppose you are given a circle. Give a construction
to find its centre.
[T-II (2011)]
19. Prove that equal chords of a circle subtend equal
angles at the centre.
[T-II (2011)]
20. 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 major arc.
[T-II (2011)]
21. In the figure, AB and CD are two equal chords of
a circle with centre O. OP and OQ are
perpendiculars on chords AB and CD respectively.
If ∠POQ = 150°, find ∠APQ.
[T-II (2011)]
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14. In the figure, ABCD is a cyclic quadrilateral. AE
is drawn parallel to CD and BA is produced up
to F. If ∠ABC = 92°, ∠FAE = 20°, find ∠BCD.
[T-II (2011)]
22. In the figure, chord AB of circle with centre O is
produced to C such that BC = OB. CO is joined
and produced to meet the circle in D. If
∠ACD = y and ∠AOD = x, show that x = 3y.
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15. ABDC is a cyclic quadrilateral and AB = AC. If
∠ACB = 70°, find ∠BDC.
[T-II (2011)]
[T-II (2011)]
A
23. O is the centre of a circle and ∠BOA = 90°, ∠COA
= 110°. Find the measure of ∠BAC. [T-II (2011)]
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16. In the figure, OA = OB = OC. Show that
∠x + ∠y = 2(∠z + ∠t).
[T-II (2011)]
24. If O is centre of circle shown in the figure and
∠AOB = 110°, then find ∠BCD. [T-II (2011)]
17. In the figure, l is a line intersecting two concentric
circles with centre P at points A, C, D and B.
Show that AC = DB.
[T-II (2011)]
12
31. In the figure, ∠ABC = 45°. Prove that OA ⊥ OC.
[T-II (2011)]
25. Prove that the line drawn through the centre of a
circle to bisect a chord is perpendicular to the
chord.
[T-II (2011)]
26. Two circles intersect at two points A and B. AD
and AC are diameters to the two circles. Prove
that B lies on the line segment DC. [T-II (2011)]
27. In the figure, O is the centre of the circle. The
angle subtended by arc ABC at the centre is 140°.
AB is produced to P. Determine ∠ADC and ∠CBP.
[T-II (2011)]
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∠BAC = 60°. Find the value of x. [T-II (2011)]
32. In a cyclic quadrilateral ABCD, if AB | | CD and
∠B = 70°, find the measures of the remaining
angles of the quadrilateral.
[T-II (2011)]
33. Find the length of the chord, which is at a distance
of 3 cm from the centre of a circle of radius
5 cm.
[T-II (2011)]
34. Find the radius of a chord, which is at a distance
of 4 cm from the centre of a circle whose radius
is 5 cm.
[T-II (2011)]
35. In the figure, O is the centre of the circle. If ∠OAC
= 35° and ∠OBC = 40°, find the value of x.
[T-II (2011)]
36. O is the circumcentre of the ∆ABC and D is the
mid point of the base BC. Prove that
∠BOD = ∠A.
[T-II (2011)]
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29. In the figure, ABCD is a cyclic quadrilateral and
∠ABC = 85°. Find the measure of ∠ADE.
[T-II (2011)]
37. Prove that any cyclic parallelogram is a rectangle.
[T-II (2011)]
38. AOB is a diameter of the circle and C, D, E are
any three points on the semicircle. Find the value
of ∠ACD + ∠BED.
[T-II (2011)]
30. In the figure, O is the centre of the circle, OM ⊥
BC, OL ⊥ AB, ON ⊥ AC and OM = ON = OL.
Is ΔABC equilateral? Give reasons.[T-II (2011)]
13
∠BCD = 120° and ∠ABD = 50°, find ∠ADB.
[T-II (2011)]
39. Two congruent circles intersect each other at points
A and B. Through A a line segment PAQ is drawn
so that P and Q lie on the two circles. Prove that
BP = BQ.
[T-II (2011)]
40. Two parallel chords of a circle whose diameter is
13 cm are respectively 5 cm and 12 cm. Find the
distance between them if they lie on opposite sides
of centre.
[T-II (2011)]
41. In the figure, ABCD is a cyclic quadrilateral. If
SHORT ANSWER TYPE QUESTIONS
[3 Marks]
9. On a common hypotenuse AB, two right triangles
ACB and ADB are situated on opposite sides.
Prove that ∠BAC = ∠BDC.
1
arc BYC. Find ∠BOC.
2
10. In the figure, O is the centre of the circle. Calculate
∠APC and ∠AOC.
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arc AXB =
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1. If the perpendicular bisector of a chord AB of a
circle PXAQBY intersects the circle at P and Q,
prove that arc PXA ≅ arc PYB.
2. In the figure, AOC is a diameter of the circle and
3. A quadrilateral ABCD is inscribed in a circle such
that AB is a diameter and ∠ADC = 130°. Find
∠BAC.
4. If two sides of a cyclic quadrilateral are parallel,
prove that remaining two sides are equal and both
diagonals are equal.
5. If a pair of opposite sides of cyclic quadrilateral
are equal, prove that its diagonals are also equal.
6. AB and AC are two equal chords of a circle. Prove
that the bisector of the angle BAC passes through
the centre of the circle.
7. B is a point on the minor arc AC of a circle with
centre D. ∠BAC = x° and ∠ADC = y°. Find the
values of x and y if ABCD is a parallelogram.
8. If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that
the quadrilateral so formed is cyclic.
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11. In the figure, AB and BC are two chords of a
circle whose centre is O such that ∠ABO =
∠CBO. Show that AB = CB.
12. Two circles with centres O and O′ intersect at
points A and B. A line PQ is drawn parallel to
O O′ through A (or B) intersecting the circles at P
and Q. Prove that PQ = 2OO′.
1. An equilateral triangle of side 9 cm is inscribed in
a circle. Find its radius.
[T-II (2011)]
2. Prove that a cyclic trapezium is always an isosceles trapezium.
[T-II (2011)]
3. If two circles intersect at the two points, prove
that their centers lie on the perpendicular bisector
of the common chord.
[T-II (2011)]
4. In the given figure, AB is a diameter of the circle;
CD is a chord equal to the radius of the circle. AC
and BD when extended intersect at a point E. Prove
that ∠AEB = 60°.
[T-II (2011)]
14
8. Two circles of radii 10 cm and 8 cm intersect and
the length of the common chord is 12 cm. Find
the distance between their centres. [T-II (2011)]
9. In the figure, O is the centre of the circle of
radius 5 cm. OP ⊥ AB, OQ ⊥ CD, AB || CD. If
AB = 6 cm, CD = 8 cm, determine PQ.
[T-II (2011)]
10. In the figure, ABCD is a parallelogram. The circle
through A, B and C intersects CD produced at E.
Prove that the AE = AD.
[T-II (2011)]
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5. If non parallel sides of trapezium are equal, prove
that it is cyclic.
[T-II (2011)]
6. The radius of a circle is 5 cm and the length of a
chord in the circle is 8 cm. Find the distance of
the chord from the centre of the circle.
[T-II (2011)]
7. In the figure, if ∠BAC = 60°, ∠ACB = 20°, find
∠ADC.
[T-II (2011)]
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LONG ANSWER TYPE QUESTIONS
[4 Marks]
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1. Show that two circles cannot intersect at more
than two points.
2. Show that the altitudes of a triangle are concurrent.
3. Prove that angle bisector of any angle of a triangle
and the perpendicular bisector of the opposite side
if intersect, they will intersect on the circumcircle
of the triangle.
4. If bisectors of opposite angles of a cyclic
quadrilateral ABCD intersect the circle,
circumscribing it at the points P and Q, prove that
PQ is a diameter of the circle.
5. ABC is an equilateral triangle inscribed in a circle
and P be any point on the minor arc BC which
does not coincide with B or C. Prove that PA is
angle bisector of ∠BPC.
BD = OD and CD ⊥ AB, find ∠CAB.
7. Prove that the angles in a segment greater than a
semi-circle is less than a right angle.
1. If two intersecting chords of a circle make equal
angles with the diameter passing through their point
of intersection. Prove that the chords are equal.
[T-II (2011)]
2. AB and AC are equal chords of a circle with centre
at O. Show that AO is perpendicular bisector of
BC.
[T-II (2011)]
15
3. AB and AC are two chords of a circle of radius r
units. If AB = 2AC, and the length of the
perpendicular from the centre on these chords are
a and b respectively, prove that 4b2 = a2 + 3r2.
[T-II (2011)]
9. In the figure, ABCD is a cyclic quadrilateral in
which AB is produced to F and BE || DC. If ∠FBE
= 20° and ∠DAB = 95°, find ∠ADC.
[T-II (2011)]
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4. P is the centre of the circle. Prove that
∠XPZ = 2 [∠XYZ + ∠XZY].
[T-II (2011)]
10. In a circle of radius 5 cm, AB and AC are two
chords such that AB = AC = 6 cm. Find the length
of chord BC.
[T-II (2011)]
11. In the figure, B and E are points on the line
segment AC and DF respectively. Show that
AD || CF.
[T-II (2011)]
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5. Prove that the angle subtended by an arc at the
centre is double the angles subtended by it at any
point on the remaining part of the circle.
[T-II (2011)]
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6. If the diagonals of a cyclic quadrilateral are
diameters of the circle through the vertices of the
quadrilateral, prove that it is a rectangle.
[T-II (2011)]
12. If O is the centre of a circle as shown in the given
figure, then prove that x + y = z. [T-II (2011)]
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7. In the figure, ∠ACE = 36°, ∠CAE = 41°. Find x,
y and z.
[T-II (2011)]
13. Prove that the quadrilateral formed (if possible)
by the internal angle bisectors of any quadrilateral
is cyclic.
[T-II (2011)]
14. Two equal chords AB and CD of a circle when
produced, intersect at the point P. Prove that
PB = PD.
[T-II (2011)]
8. In the figure, ABCD is a cyclic quadrilateral, O is
the centre of the circle. If ∠BOD = 160°, find
∠BPD.
[T-II (2011)]
16
15. In the figure, equal chords AB and CD intersect
each other at Q at right angle. P and R are mid
points of AB and CD respectively. Show that
OPQR is a square.
[T-II (2011)]
17. In the figure, find the values of a, b, c and d.
Given ∠BCD = 43° and ∠BAE = 62°.
[T-II (2011)]
19. Prove that the circle drawn on any one of the
equal sides of an isosceles triangle as diameter
bisects the base of the triangle.
[T-II (2011)]
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16. Prove that line joining the centers of two
intersecting circles subtends equal angles at the
two points of intersection of circles. [T-II (2011)]
18. Two circles intersect at two points B and C.
Through B, two line segments ABD and PBQ are
drawn to intersect the circles at A, D and P, Q
respectively. Prove that ∠ACP = ∠QCD.
[T-II (2011)]
FORMATIVE ASSESSMENT
Activity-1
Objective : To verify that the angle subtended by an arc at the centre of a circle is twice the angle subtended by
the same arc at any other point on the remaining part of the circle, using the method of paper cutting,
pasting and folding.
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Materials Required : White sheets of paper, tracing paper, a pair of scissors, gluestick, colour pencils, geometry
box, etc.
Procedure :
1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Mark two points A and
B on the boundary of the circle to get arc AB. Colour the minor arc AB green.
Figure-1
17
2. Take any point P on the remaining part of the circle. Join OA, OB, PA and PB.
Figure-2
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3. Make two replicas of ∠APB using tracing paper. Shade the angles using different colours.
Figure-3
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4. Paste the two replicas of ∠APB adjacent to each other on ∠AOB as shown in the figure.
Figure-4
Observations :
1. In figure 2, ∠AOB is the angle subtended by arc AB at the centre and ∠APB is the angle subtended by arc
AB on the remaining part of the circle.
2. In figure 3, each angle is a replica of ∠APB.
3. In figure 4, we see that the two replicas of ∠APB completely cover the angle AOB.
So, ∠AOB = 2∠APB.
Conclusion : From the above activity, it is verified that the angle subtended by an arc at the centre of a circle is
twice the angle subtended by the same arc at any other point on the remaining part of the circle.
Do Yourself : Verify the above property by taking three circles of different radii.
18
Activity-2
Objective : To verify that the angles in the same segment of a circle are equal, using the method of paper cutting,
pasting and folding.
Materials Required : White sheets of paper, tracing paper, a pair of scissors, gluestick, colour pencils, geometry box,
etc.
Procedure :
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1. On a white sheet of paper, draw a circle of any convenient radius. Draw a chord AB of the circle.
Figure-1
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2. Take any three points P, Q and R on the major arc AB of the circle. Join A to P, B to P, A to Q, B to Q, A to R
and B to R.
Figure-2
3. On a tracing paper, trace each of the angles APB, AQB and ARB. Shade the traced copies using different
colours.
Figure-3
19
4. Place the three cut outs one over the other such that the vertices P, Q and R coincide and PA, QA and RA fall
along the same direction.
Figure-4
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Observations :
1. In figure 2, ∠APB, ∠AQB and ∠ARB are the angles in the same major segment AB.
2. In figure 4, we see that ∠APB, ∠AQB and ∠ARB coincide.
So, ∠APB = ∠AQB = ∠ARB
Conclusion : From the above activity, it is verified that the angles in the same segment of a circle are equal.
Do Yourself : Verify the above property by taking three circles of different radii.
Activity-3
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Objective : To verify using the method of paper cuting, pasting and folding that
(a) the angle in a semi circle is a right angle
(b) the angle in a major segment is acute
(c) the angle in a minor segment is obtuse.
Materials Required : White sheets of paper, tracing paper, cut out of a right angle, colour pencils, a pair of scissors,
gluestick, geometry box, etc.
Procedure : (a) To verify that the angle in a semicircle is a right angle :
1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw its diameter AB as
shown.
Figure-1
2. Take any point P on the semicircle. Join A to P and B to P.
Figure-2
20
3. Make two replicas of ∠APB on tracing paper. Shade the replicas using different colours.
Figure-3
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4. On a white sheet of paper, draw a straight line XY. Paste the replicas obtained in figure 3 on
XY and adjacent to each other such that AP and BP coincide as shown in the figure.
Figure-4
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(b) To verify that the angle in a major segment is acute :
1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw a chord AB which does
not pass through O.
Figure 5
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2. Take any point P on the major segment. Join P to A and P to B.
Figure-6
3. Trace ∠APB on a tracing paper.
Figure-7
21
4. Paste the traced copy of ∠APB on the cut out of a right angled triangle XYZ, right-angled at Y such that PA
falls along YZ.
Figure-8
(c) To verify that the angle in a minor segment is obtuse :
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1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw any chord AB which
does not pass through O.
Figure-9
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2. Take any point P on the minor segment. Join P to A and P to B.
Figure-10
3. Trace ∠APB on a tracing paper.
Figure-11
4. Paste the traced copy of ∠APB on the cut out of a right-angled triangle XYZ, right angled at Y, such that PA
falls along YZ.
22
Figure-12
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Observations :
1. In figure 2, APB is a semicircle. So, ∠APB is an angle in a semicircle.
2. In figure 4, we see that PB and PA fall along XY.
Or ∠APB + ∠APB = a straight angle = 180°
⇒ 2∠APB = 180°
⇒ ∠APB = 90°
Hence, angle in a semicircle is a right angle.
3. In figure 7, ∠APB is an angle formed in the major segment of a circle.
4. In figure 8, we see that the side PB of ∠APB lies to the right of XY of ∠XYZ, ie, ∠APB is less than a right
angle, or ∠ΑPB is acute.
Hence, the angle in a major segment is acute.
5. In figure 11, ∠APB is an angle formed in the minor segment of a circle.
6. In figure 12, we see that the side PB of ∠PAB lies to the left of XY of ∠XYZ ie, ∠APB is greater than ∠XYZ
or ∠APB is obtuse.
Hence, the angle in a minor segment is obtuse.
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Conclusion : From the above activity, it is verified that :
(a) the angle in a semicircle is a right angle.
(b) the angle in a major segment is acute.
(c) the angle in a minor segment is obtuse.
Activity-4
Objective : To verify using the method of paper cutting, pasting and folding that
(a) the sum of either pair of opposite angles of a cyclic quadrilateral is 180°
(b) in a cyclic quadrilateral the exterior angle is equal to the interior opposite angle.
Materials Required : White sheets of paper, tracing paper, colour pencils, a pair of scissors, gluestick, geometry box, etc.
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Procedure :
(a) 1. On a white sheet of paper, draw a circle of any convenient radius. Mark four points P, Q, R, S on the circumference
of the circle. Join P to Q, Q to R, R to S and S to P.
Figure-1
23
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2. Colour the quadrilateral PQRS as shown in the figure and cut it into four parts such that each part contains one
angle, ie, ∠P, ∠Q, ∠R and ∠S.
Figure-2
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3. On a white sheet of paper, paste ∠P and ∠R adjacent to each other. Similarly, paste ∠Q and ∠S adjacent to
each other.
Figure-3
(b) 1. Repeat step 1 of part (a).
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2. Extend PQ to PT to form an exterior angle RQT. Shade ∠RQT.
Figure-4
3. Trace ∠PSR on a tracing paper and colour it.
Figure-5
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4. Paste the traced copy of ∠PSR on ∠RQT such that S falls at Q and SP falls along QT.
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Figure-6
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Observations :
1. In figure 2, ∠P, ∠Q, ∠R and ∠S are the four angles of the cyclic quadrilateral PQRS.
2. In figure 3(a), we see that ∠R and ∠P form a straight angle and in figure 3(b), ∠Q and ∠S form a straight
angle.
So, ∠P + ∠R = 180° and ∠Q + ∠S = 180°.
Hence, the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
3. In figure 5, ∠PSR is the angle opposite to the exterior angle RQT.
4. In figure 6, we see that ∠PSR completely covers ∠TQR.
Hence, in a cyclic quadrilateral the exterior angle is equal to the interior opposite angle.
Conclusion : From the above activity, it is verified that
(a) the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
(b) in a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
25

sa2 ix maths(ch 10)

Transcription

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