Table of Contents
NCERT solutions for Class 9 provide comprehensive and structured explanations to help students grasp the concepts of circles with ease. The circle chapter in Class 9 Math introduces fundamental concepts like the definition of a circle, its properties, and theorems related to angles subtended by arcs. The NCERT solutions not only assist students in understanding the theoretical aspects but also offer step-by-step solutions to a variety of problems. By utilizing these solutions, students can enhance their problem-solving skills and build a strong foundation in geometry. The carefully crafted content of NCERT books for Class 9 ensures that students gain a solid understanding of circle-related topics, making mathematics an enjoyable and accessible subject for them. Also Check out our CBSE Notes for Class 9
The topics and sub-topics in NCERT Class 9 Maths Textbook Chapter 10 Circles
- Circles
- Introduction
- Circles And Its Related Terms: A Review
- Angle Subtended By A Chord At A Point
- Perpendicular From The Centre To A Chord
- Circle Through Three Points
- Equal Chords And Their Distances From The Centre
- Angle Subtended By An Arc Of A Circle
- Cyclic Quadrilaterals
- Summary
NCERT Solutions for Class 9 All Subjects
- NCERT Solutions for Class 9 Maths
- NCERT Solutions for Class 9 Science
- NCERT Solutions for Class 9 Social Science
- NCERT Solutions for Class 9 English
IMPORTANT QUESTIONS
VERY SHORT ANSWER TYPE QUESTIONS
1. 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. [CBSE-14-17DIG1U]
Answer.
2. In the given figure, what is the measure of angle x ?
Answer. We know that exterior angle of a cyclic quadrilateral is equal to interior opposite angle.
.-. ∠CBE = ∠ADC
=> x = 120°
Also Check: CBSE Notes Class 9 Maths
3. In the given figure, if O is the centre of circle and ∠POQ = 110°, then find ∠PRQ
Answer. We know that angle subtended by an arc at the centre is double the angle subtended by it at the remaining part of the circle.
∠PRQ =1/2 ∠POQ
= 1/2 x 110°
= 55°
4. In the given figure, ΔABC is an equilateral triangle and ABDC is a cyclic quadrilateral, then find the measure of ∠BDC.
Answer.
5. In the given figure, O is the centre of the circle. PQ is a chord of the circle and R is any point on the circle. If ∠PRQ = l and ∠OPQ = m, then find l + m.
Answer.
6. The given figure shows a circle with centre O in which a diameter AB bisects the chord PQ at the point R. If PR = RQ = 8 cm and RB = 4 cm, then find the radius of the circle.
Answer.
7. In the given figure, ABCD is a cyclic quadrilateral such that ∠ADB= 40° and ∠DCA = 70°, then find the measure of ∠DAB .
Answer.
8. In the figure, ‘O’ is the centre of the circle, ∠ABO= 20° and ∠ACO= 30°, where A, B, C are points on the circle. What is the value of x ?
Answer.
9. In the given figure, if O is the centre of circle. Chord AB is equal to radius of the circle, then find ∠ACB.
Answer.
10. In the given figure, if ∠OAB = 40°, then find the measure of ∠ACB. [NCERT Exemplar Problem]
Answer.
11. In the given figure, O is the centre of the circle. If∠BOC= 120°, then find the value of x.
Answer.
12. In the given figure, O is the centre of the circle, then compare the chords.
Answer. In chords AB and CD, AB is passing through the centre of the circle. AB is the diameter of circle. Thus, AB>CD [v diameter is the largest chord]
13. In the given figure, ∠ACP = 40° and ∠BPD = 120°, then find ∠CBD .
Answer.
14. In the given figure, if ∠SEC = 120°, ∠DCE = 25°, then find ∠BAC.
Answer.
15. AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, then find the distance of AB from the centre of the circle. [CBSE March 2012]
Answer.
16. In the given figure, if ∠DAB = 60°, ∠ABD= 50°, then find ∠ACB. [NCERT Exemplar Problem]
Answer.
17. In the given figure, AB || DC. If ∠A = 50°, then find the measure of ∠ABC. [NCERT Exemplar Problem]
Answer.
SHORT ANSWER QUESTIONS TYPE-I
18. Equal chords of a circle subtend equal angles at the centre. [CBSE March 2012]
Answer.
19. If the angles subtended by the chords of a circle at the centre are equal, then chords are equal. [CBSE March 2012]
Answer.
20. In the figure, O is the centre of the circle and ∠ABC= 45°. Show that OA⊥OC[/latex]. [CBSE March 2013]
Answer. Since angle subtended by an arc at the centre of the circle is double the angle subtended it at any point on the remaining part of the circle.
21. In the given figure, ∠PQR= 100°, where P,Q and R are points on a circle with centre O. Find ∠OPR. [CBSE March 2012]
Answer.
22. In figure, ABCD is a cyclic quadrilateral in which AB is extended till F and BE || DC. If ∠FBE= 20° and ∠DAB = 95°, then find∠ADC. [CBSE March 2012]
Answer.
23. 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. [CBSE March 2011 ]
Answer.
24. 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 (see fig.). [CBSE March 2013]
Answer. Draw OM⊥ l
Since perpendicular from the centre of a circle to a chord of the circle bisects the chord.
BM = CM …(i)
and AM = DM …(ii)
Subtracting (i) from (ii), we have
AM – BM = DM – CM
AB = CD
25. In the given figure, P is the centre of the circle. Prove that : ∠ XPZ = 2(∠ XZY +∠ YXZ)
Answer.
26. 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 (see fig.). Prove that ∠ ACP =∠ QCD.
Answer.
27. If the diagonals of a cyclic quadrilateral are diameters of the circle through the opposite vertices of the quadrilateral. Prove that the quadrilateral is a rectangle. [CBSE-14-GDQNI3W]
Answer.
SHORT ANSWER QUESTIONS TYPE-II
28. If the non-parallel sides of a trapezium are equal, prove that it is cyclic. [CBSE March 2013]
Answer.
29. ABCD is a parallelogram. The circle through A, B and C intersects (produce if necessary) at E. Prove that AE = AD.
Answer.
30. ABCD is a cyclic quadrilateral in which AB and CD when produced meet in E and EA = ED. Prove that : (i) AD\\BC (ii) EB = EC
Answer.
31. If two equal chords of a circle intersect within a circle, prove that the line segment joining the point of intersection to the centre makes equal angles with the chords. [CBSE-15-NS72LP7]
Answer.
32. Two circles whose centres are O and O’ intersect at P. Through P, a line parallel to OO’, intersecting the circles at C and D is drawn as shown in the figure. Prove the CD = 200′. [CBSE-15-6DWMW5A] [CBSE-14-ERFKZ8H]
Answer.
33. If 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 . [NCERT Exemplar Problem]
Answer.
34. In the given 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 = 120°, find ∠APQ . [CBSE-14-ERFKZ8H]
Answer.
LONG ANSWER TYPE QUESTIONS
35. Show that the quadrilateral formed by angle bisectors of a cyclic quadrilateral, is also cyclic. [CBSE March 2012]
Answer.
36. Prove that 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. [CBSE March 2012]
Answer.
37. Prove that the angle bisectors of the angle formed by producing opposite sides of a cyclic quadrilateral (provided they are not parallel) intersect at right angle.
Answer.
38. Bisectors of angles A, B and C of triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the ΔDEF are 90° -∠ A/2 ,90° – ∠ B/2 and 90°- ∠ C/2 respectively.
Answer.
39. In the given figure, O is the centre of a circle of radius r cm, OP and OQ are perpendiculars to AB and CD respectively and PQ = 1 cm. If AB II CD, AB = 6cm and CD = 8cm, determiner. [ CBSE-15-6DWMW5A]
Answer. Since the perpendicular drawn from the centre of the circle to a chord bisects the chord. Therefore, P and Q are mid-points of AB and CD respectively.
40. If two chords AB and CD of a circle AYDZBWCX intersect at right angles (see fig.), prove that arc CXA + arc DZB = arc AYD + arc BWC = semicircle. [NCERT Exemplar Problem]
Answer. Given two chords AB and CD of a circle intersect at right angle. Let P be the point of intersection of the chord and O be the centre of circle AYDZBWCX.
41. In the given figure, AC is a diameter of the circle with centre O. Chord BD is perpendicular to AC. Write down the measures of angles a, b, c and d in terms of x. [CBSE-15-NS72LP7]
Answer.
42. The bisector of \(\angle B\) of an isosceles triangle ABC with AB = AC meets the circumcircle of ΔABC at P as shown in the given figure. If AP and BC produced meet at Q, prove that CQ = CA. [CBSE-14-GDQNI3W]
Answer.
Value Based Questions
1. A small cottage industry employing people from a nearby slum area prepares round table cloths having six equal designs in the six segment formed by equal chords AB, BC, CD, DE, EF and FA. If O is the centre of round table cloth (see figure). Find \(\angle AOB\), \(\angle AEB \) and \(\angle AFB\).What value is depicted through this question ? [CBSE-14-17DJG1U]
Answer.
Value depicted : By employing people from a slum area to prepare round table cloths realize their social responsibility to work for helping the ones in need.
2. Three students Priyanka, Sania and David are protesting against killing innocent animals for commercial purposes in a circular park of radius 20 m. They are standing at equal distances on its boundary by holding banners in their hands.
(i) Find the distance between each of them.
(ii) Which mathematical concept is used in it ?
(iii) How does an act like this reflects their attitude towards society ?
Answer.
3. A circular park of radius 10 m is situated in a colony. Three students Ashok, Raman and Kanaihya are standing at equal distances on its circumference each having a toy telephone in his hands to talk each other about Honesty, Peace and Discipline.
(i) Find the length of the string of each phone.
(ii) Write the role of discipline in students’ life.
Answer. Let us assume, A, B and C be the position of three students Ashok, Raman and Kanaihya respectively on the circumference of the circular park with centre O and radius 10 m. Since the centre of circle coincides with the centroid of the equilateral ΔABC.
4. Three scouts Rajat, Rohit and Ramit in the cultural show holded three stringed balloons with a message ‘Stop Child Labour’. Keeping themselves on the boundary of a circle of radius 25 cm, each scout holded the string tightly. Find the distance between Rajat and Ramit, when distance between Rajat and Rohit and Rohit and Ramit is 30 cm. What message was given by scouts and why ?
Answer.
