PODAR INTERNATIONAL SCHOOL

Subject: Mathematics Chapter 9: Circles

Std.: IX Answer Scheme Practice Sheet

Section A: Multiple Choice Questions (1 mark each)

Q1. Which of the following is not a chord?
(a) Diameter (b) Radius (c) Secant (d) Bisector
Answer: option (b)
Q2. Given alongside is a circle with center O
Which of the following is not true about the given figure?

a. MGN is the minor sector

b. MN is the minor arc
c. MDN is the major arc
d. AR is a chord
Answer: option (a)
Q3. The longest chord of a circle is the
(a) Sector (b) Segment (c) Radii (d) Diameter
Answer: option (d)
Q4. The length of a chord of circle of radius 10 cm is 12 cm. Determine the distance of the
chord from the centre
(a) 8 cm (b) 7 cm (c) 6cm (d) 5cm

Q5. The length of a chord of circle is 4 cm. If its perpendicular distance from the centre is
1.5 cm, determine the radius of the circle
a. 2.5 cm b. 1.5 cm c. 6 cm d. 5 cm
Answer: option (a)
Q6. The line drawn through the centre of a circle to ______ a chord is perpendicular to the chord.
(a) Trisect (b) Bisect (c) Coincide (d) None of these
Answer: option (b)
Q7. From the diagram given below, angle subtended by the chord AB at
C
the centre is __________

A B

(a) ∠AOB (b) ∠ACO (c) ∠ACB (d) ∠OAB

Answer: option (a)
Q8. Find the length of a chord, which is at a distance of 24 cm from the centre of a circle
whose diameter is 50 cm.

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 (a) 12 cm (b) 14 cm (c) 15 cm (d) 16 cm
Answer: option (b)
Q9. From the diagram given below, if O is the centre of the circle and OL
is perpendicular to chord AB, then which of the following is true?
O

A L B
(a) AL> LB (b) AL < LB (c) AL = LB (d) AL ≠ LB
Answer: option (a)
Q10. In the figure, diameter AB and a chord AC have a common end point
A. If the length of AB is 20 cm and of AC is 12 cm, how far is AC from A D C
the centre of the circle?
O
B
(a) 10 cm (b) 7 cm (c) 8 cm (d) 12 cm
Answer: option (c)
Q11. If a line intersects two concentric circles at A, B C and D, then

(a) AB = CD (b) AB = BC (c) AB > CD (d) AB < CD

Answer: option (a)
Q12. 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.
(a) 150° (b) 60° (c) 30° (d) 120°
Answer: option (c)
Q13. Given below is a circle with diameter 20 cm. What is the
distance between the chords? How much this distance will
change if the chords are in the same direction of the centre?

(a) 14 cm; it will increase by 12 cm

(b) 7 cm; it will remain same
(c) 14 cm; it will decrease by 12 cm
(d) 2 cm; it will increase by 12 cm
Answer: option (c)
Q14. In the figure, if AOB is the diameter of the circle and AC = BC, then
C
∠CAB = ?

A B
O

(a) 30° (b) 60° (c) 90° (d) 45°

Answer: option (d)

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Q15. In the given figure, the value of x is
A
D
x

40°O
B C

(a) 100° (b) 80° (c) 140° (d) 40°

Answer: option (c)
Q16. In the given figure the measure of ∠BOD = 88°. Find the
measure of ∠BAC, given O is the centre of the circle. B

A O

(a) 44° (b) 60° (c) 88° (d) 176°

Answer: option (a)
Q17. PQRS is a cyclic quadrilateral such that PQ is a diameter of the circle circumscribing the
quadrilateral and ∠PSR = 150° then the measure of ∠QPR is
(a) 60° (b) 50° (c) 40° (d) 30°
Answer: option (a)
Q18. In the given circle, if BD = DC, and BCD= 25° , then what is
BAC ?

(a) 25° (b) 50° (c) 80° (d) 100°

Answer: option (b)
Direction for questions 19 & 20: In question numbers 19 and 20, a statement of Assertion (A)
is followed by a statement of Reason (R). Choose the correct option.
Q19. Assertion (A): In the given figure, ∠ABC = 69° and ∠ACB =
31°. Then, ∠BDC = 80°.

Reason (R): Angles in the same segment of a circle are equal.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of
Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation
of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.

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 (d) Assertion (A) is false but Reason (R) is true.
Answer: option (a)

Q20. Assertion (A): In an isosceles triangle ABC with AB = AC, a circle passes through B and C
intersects the sides AB and AC at D and E respectively, then DE || BC.
Reason (R): Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of
that quadrilateral.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of
Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation
of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
Answer: option (a)

Section B: Short Answer Questions [Type I] (2 marks each)

Q1. If BC is a diameter of the circle and ∠BAO = 60°. Then find the
value of ∠ADC.

Sol: 60°
Q2. In Figure, ∠ ∠ ABC = 69°, ACB = 31°, find ∠BDC.
D
A

69 31
B C
° °

Sol: 80°
Q3. In the given figure,, O is the centre of a circle.
Determine (a) AEC (b) reflex AOC

Sol: 68°, 224°

Q4. Find the length of a chord which is at a distance of 12 cm from the centre of the circle of radius
13 cm.

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Sol: 10 cm
Q5. AB and CD are equal chords of a circle whose centre is O. If C A
OM ⊥ AB and ON ⊥ CD, prove that ∠OMN = ∠ONM. M N
B D
O

Sol: ----
Q6. In the given figure, O is the centre of the circle. Quadrilateral S T
PQTS is a cyclice quadrilateral. If ∠POT = 130°, find ∠PST. x°
130°
P Q
O

Sol: ∠PST =115°

Q7. In the given figure, O is the centre of the circle with chords
AP and BP being produced to R and Q respectively. If ∠QPR
= 35°, find the measure of ∠AOB.

Sol: 70°
Q8. In the given figure, ΔABC is an equilateral triangle and ABDC is a
cyclic quadrilateral, then find the measure of ∠BDC.

Sol: 120°
Q9. In the given figure, ABCD is a cyclic quadrilateral such that A D
∠ADB= 40° and ∠DCA = 70°, then find the measure of ∠DAB.
40°
70°
B C

Sol: 70°
Q10. Bisector AD of ∠BAC of ΔABC passes through the centre O of the circumcircle of ΔABC.
Prove that AB = AC
Sol: ----
Section B: Short Answer Questions [Type II] (3 marks each)
Q1. Prove that the angle subtended by an arc at the center is double the angle subtended by it at any
point on the remaining part of the circle.

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Sol:
Q2. In the figure, PR is a diameter of the circle with centre O, S
PR = 10 cm, PS = 8 cm and PQ = 3 cm. Calculate the perimeter of the
cyclic quadrilateral PQRS.
O R
P

Q
Sol:

PQRS = 17  91 cm 
Q3. The angle subtended by an arc BC of a circle cantered at O is
2 + 50.
(a) Find BAC in terms of .
(b) If  = 30, find the reflex angle of BOC
(c) If the length of the chord of the circle AB = 16 cm and is at the
distance of 15 cm from the centre of the circle, then find the
radius of the circle

Sol: (a)  + 25° (b) 250° (c) 17 cm

Q4. Prove that equal chords of a circle subtend equal angles at the centre.

Sol: ---
Q5. In the adjoining figure, ED is a chord parallel to the diameter
AC of the circle ABCDE. If ∠CBE = 63o, calculate ∠DEC.

Sol: ∠DEC = 27°

Q6. Two chords AB and CD of lengths 8 cm and 6 cm of a circle are parallel and are on the same
side of its centre. If the distance between them is 1 cm, find the radius of the circle.
Sol: 5 cm
Q7. In the adjoining figure, O is the center of the circle.
If ∠BCO = 30o, find x and y.

Sol: x = 30, y = 15

Q8. PQRS is aparallelogram. The circle through P, Q and R intersects RS produced at T. Prove that
PS = PT.
Sol: ---
Q9. Two chords AB and AC of a circle subtend angles equal to 90° and 150° respectively at the

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 centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.
Sol: ---
Sol:
Section D: Long Answer Questions (5 marks each)
Q1. AB and AC are two chords of a circle of radius r such that AB = 2AC.
If p and q are the distances AB and AC from the centre, prove that C
4q 2  p 2  3r 2 M q O
r p
A L B

Sol: --
Q2. If two circles intersect at two points, prove that theline through the centres is the perpendicular
bisector of the common chord.
Sol: ----
Q3. In the given figure, QR is the bisector of SQT where PQRS
is a cyclic quadrilateral. Prove that SR = PR. Q T
P

S R

Sol: ---
Q4. ABCDE is a pentagon inscribed in a semicircle with centre O. Find
the numerical value of ∠ABC + ∠CDE.

Sol: 270°
Q5. The lengths of two parallel chords on the same side of the centre are 6 cm and 8 cm. If the
distance between them is 1 cm from the centre, find the radius of the circle.
Sol: 5 cm
Section E: Case Study Based Questions
Case Study I
I There is a well that serves the drinking water needs of a village. The top of the well is circular
with radius 5 feet. Two rods of length 6 feet each are situated on the top of the well for the
purpose of pulling bucket containing water with the help of two pulleys. These two rods are
attached as shown in the given figure. Based on the information, answer the following
questions

O
5m 5m
B C
6m 6m

Q1. Find the length of the line segment joining the point O and the point A.

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Sol: 5m
Q2. What will be the measure of the angle ODB formed at the point D.
Sol: 90°
Q3. Find the length of the line segment joining the point B and point C.
OR
If OA intersects BC at D as shown in the figure, then find the length of the line segment joining
B and D.
Sol: 9.6 m
OR
4.8 m

Case Study II
Three friends are standing in a circular park where they observe three shops situated at
P, Q, R as shown in the figure. The distance between shop P and Q is 8 m, distance
between shop Q and R is 10 m and between shop P and R is 6 m. Considering O as the
centre of the circle, answer the following questions.
P

Q R
O

Q1. Find the radius of the circle.

Sol: 5m
Q2. Find the measure of ∠QPR
Sol: 90°
Q3. Find the length of the perpendicular OT, drawn from the centre O to the chord PQ
OR
Find the length of the perpendicular OS drawn from the centre O to the chord PR.
Sol: 3m
OR
4m

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