Circles — Class 9 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Circles" — 8 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Circles" — 8 important questions with detailed answers for CBSE board exam preparat…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

Define a circle and state the following properties: (1) All radii of a circle…
State the theorem about angles subtended by an arc at the center and on the c…
In circle with center O and radius 5 cm, a chord AB is at a perpendicular dis…
State the theorem about angles in the same segment: Two angles subtended by t…
In a circle, two chords AB and CD intersect at point P inside the circle. If …
A chord of a circle subtends angles of 60° at two points on the major arc and…
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
Define a circle and state the following properties: (1) A…	✓ Solved
State the theorem about angles subtended by an arc at the…	✓ Solved
In circle with center O and radius 5 cm, a chord AB is at…	✓ Solved
State the theorem about angles in the same segment: Two a…	✓ Solved
In a circle, two chords AB and CD intersect at point P in…	✓ Solved
A chord of a circle subtends angles of 60° at two points …	✓ Solved

Showing 6 of 8 questions

Q1: Define a circle and state the following properties: (1) All radii of a circle are equal. (2) A chord is a line segment joining any two points on the circle. (3) The longest chord is the diameter. If a circle has radius 7 cm, find the diameter and the circumference.

Step 1: Definition of Circle: A circle is the locus of all points in a plane that are at a constant distance (radius) from a fixed point (center). Step 2: Properties: (1) All radii of a circle are equal: Since all points on the circle are equidistant from the center, all radii r are equal. (2) Chord: A line segment with both endpoints on the circle. Examples: diameter (through center), any other line segment between two points on circle. (3) Diameter is the longest chord: Diameter = 2r (passes ...

Q2: State the theorem about angles subtended by an arc at the center and on the circumference: If an arc AB subtends an angle of 60° at the center O, what angle does the same arc subtend at a point C on the circumference (on the major arc)?

Step 1: Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circle (on the major arc). Angle at center = 2 × Angle at circumference Step 2: Given: Arc AB subtends angle AOB = 60° at center O Point C is on the major arc AB Find: angle ACB (angle at circumference) Step 3: Apply the theorem: angle AOB = 2 × angle ACB 60° = 2 × angle ACB angle ACB = 60°/2 angle ACB = 30° Step 4: Verification: This theorem holds for all...

Q3: In circle with center O and radius 5 cm, a chord AB is at a perpendicular distance of 3 cm from the center O. Find the length of chord AB.

Step 1: Given: Circle with center O and radius r = 5 cm Chord AB at perpendicular distance d = 3 cm from center O Step 2: Let M be the foot of the perpendicular from O to chord AB. Then OM ⊥ AB and OM = 3 cm Step 3: Important property: The perpendicular from the center of a circle to a chord bisects the chord. Therefore: AM = MB = AB/2 Step 4: In right triangle OMA: angle OMA = 90° OA = 5 cm (radius, hypotenuse) OM = 3 cm (perpendicular distance) AM = ? (half of chord) Step 5: By Pythagoras ...

Q4: State the theorem about angles in the same segment: Two angles subtended by the same chord at two different points on the same arc are equal. In circle with center O, chord AB subtends angle 40° at point C on the major arc. Find the angle it subtends at point D, also on the major arc.

Step 1: Theorem (Angles in the Same Segment): Angles subtended by the same chord at different points on the same arc of a circle are equal. If chord AB subtends angles at points C and D on the same arc, then angle ACB = angle ADB. Step 2: Given: Circle with center O Chord AB Point C on major arc: angle ACB = 40° Point D also on major arc Find: angle ADB Step 3: Since C and D are both on the major arc (same segment), and they are both subtended by the same chord AB: angle ADB = angle ACB Step...

Q5: In a circle, two chords AB and CD intersect at point P inside the circle. If AP = 6 cm, PB = 4 cm, and CP = 3 cm, find the length of PD.

Step 1: Theorem (Intersecting Chords): When two chords intersect inside a circle, the products of their segments are equal. If chords AB and CD intersect at P, then: AP × PB = CP × PD Step 2: Given: Chord AB with intersection point P AP = 6 cm PB = 4 cm Chord CD with intersection point P CP = 3 cm PD = ? Step 3: Apply the intersecting chords theorem: AP × PB = CP × PD 6 × 4 = 3 × PD 24 = 3 × PD Step 4: Solve for PD: PD = 24/3 PD = 8 cm Step 5: Verification: Product of segments of chord AB =...

Q6: A chord of a circle subtends angles of 60° at two points on the major arc and 120° at the minor arc. Verify this using the angle-in-segment theorem.

Step 1: Angle-in-Segment Theorem Verification: Angles subtended by a chord at points on the major arc are equal. Angles subtended by a chord at points on the minor arc are equal. Angles in the major arc + angles in the minor arc = 180° Step 2: Given: Chord AB Angle at point C on major arc = 60° Angle at point D on minor arc = 120° Step 3: Verify the relationship: angle ACB + angle ADB = 60° + 120° = 180° ✓ Step 4: Explanation using central angle: Let the central angle AOB = θ For point C on ...

Showing 6 of 8 questions. Visit the full page for complete solutions.

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