Circles is an important and concept-based chapter in Class 9 Mathematics that focuses on the properties and theorems related to circles. This chapter covers key topics such as the circle and its elements (radius, diameter, circumference), tangents, secants, theorems related to angles subtended by a chord, tangents from an external point, and the length of a tangent, which are frequently asked in school exams.
Q.
Fill in the blanks:
(i) The centre of a circle lies in_______ of the circle. (exterior/ interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in______ of the circle. (exterior/ interior)
(iii) The longest chord of a circle is a _____ of the circle.
(iv) An arc is a_______ when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and ________of the circle.
(vi) A circle divides the plane, on which it lies, in_______ parts.
Q.
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
Q.
Q.
ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.
Q.
Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.
Q.
Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.
Q.
If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
Q.
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 the figure below). Prove that ∠ACP=∠QCD.
Q.
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
Q.
In the figure below A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.
Q.
Write True or False: Give reasons for your answers.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
Q.
In the figure below, ∠ABC = 69°, ∠ACB=31°, find ∠BDC.
Q.
In the figure below, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.
Q.
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 minor arc and also at a point on the major arc.
Q.
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
Q.
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
Q.
If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
Q.
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
Q.
If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
Q.
In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.
NCERT Solutions for Class 9 Maths Chapter 9 – Circles are prepared strictly according to the latest CBSE syllabus and exam pattern. The solutions are explained in simple, step-by-step language with clear diagrams and logical reasoning, helping students understand theorems clearly and solve problems with confidence.
Q1) Fill in the blanks:
(i) The centre of a circle lies in _______ of the circle. (exterior/ interior)
Answer: interior
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in _______ of the circle. (exterior/ interior)
Answer: exterior
(iii) The longest chord of a circle is a _______ of the circle.
Answer: diameter
(iv) An arc is a _______ when its ends are the ends of a diameter.
Answer: semi-circle
(v) Segment of a circle is the region between an arc and _______ of the circle.
Answer: chord
(vi) A circle divides the plane, on which it lies, in _______ parts.
Answer: three parts
Q2) Write True or False: Give reasons for your answers.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
Answer: True. Since, all the points on the circumference are equidistant from the centre of the circle, and this distance is called the radius of the circle.
(ii) A circle has only a finite number of equal chords.
Answer: False. Since, there are infinite points on the circumference, we can draw an infinite number of chords of the same length. Hence, a circle has an infinite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
Answer: False. Three equal arcs in a circle will not be major arcs because there is no minor arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
Answer: True. Since, a chord equal to twice the radius is called the diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
Answer: False. Sector is the region bounded by an arc and two radii of the circle. OAB is the sector of the circle in the figure.
(vi) A circle is a plane figure.
Answer: True. A circle is a two-dimensional figure as it has no thickness, so it is called a plane figure.
FAQs: Class 9 Maths Chapter 9 – Circles
Q1. Is Circles an important chapter for Class 9 exams?
Yes, it is an important geometry chapter with many theorem-based questions.
Q2. Which topics are most important in this chapter?
Tangent, secant, angle subtended by a chord, and length of a tangent.
Q3. Are proof-based questions asked in this chapter?
Yes, theorem-based and proof-based questions are common.
Q4. Are diagrams important in this chapter?
Yes, circle-related diagrams are crucial for understanding and solving problems.
Q5. How do NCERT Solutions help?
They provide NCERT-aligned, exam-ready explanations with clear diagrams and proofs.