NCERT MCQ Solutions for Class 9 Maths Chapter 9 Circles

A circle is defined as the set of all points in a plane which are equidistant from a fixed point called the centre. This constant distance from the centre to any point on the circle is called the radius and forms the basic definition of a circle.

The longer the chord of a circle the greater is the angle subtended by it at the centre. This relationship is observed by drawing different chords and measuring the central angles they subtend.

Equal chords of a circle subtend equal angles at the centre. This is proved using SSS triangle congruence and the concept of corresponding parts of congruent triangles (CPCT) which are fundamental geometric principles.

If two chords of a circle subtend equal angles at the centre then those chords are equal. This is the converse of Theorem and is proved by triangle congruence and using the CPCT rule.

The perpendicular drawn from the centre of a circle to a chord bisects the chord. This is demonstrated by proving congruent triangles formed by the radii and the perpendicular segment using RHS congruence.

The centre of a circle lies in the interior of the circle. By definition, the centre is the point from which all points on the boundary (circumference) of the circle are equidistant.

A point whose distance from the centre of a circle is greater than its radius lies in the exterior of the circle. This is because the radius defines the distance to the boundary, and any point farther than this is outside.

Theorem states that if a line drawn from the centre bisects a chord then it is perpendicular to the chord. This is verified by triangle congruence and symmetry properties of circles.

Equal chords of a circle are equidistant from the centre. This is demonstrated using paper folding and verifying through triangle congruence that equal chords fall at the same distance from the centre.

Theorem proves that chords that are equidistant from the centre of a circle are equal in length. This is proved using congruent triangles and equal perpendicular distances from the centre.

Congruent arcs subtend equal angles at the centre. This is because equal chords define equal arcs and these arcs subtend equal central angles. The relationship is discussed when exploring chord and arc symmetry.

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. This is shown using triangle and angle construction.

The longest chord of a circle is the diameter of the circle. The diameter passes through the centre of the circle and is twice the length of the radius, making it the longest possible chord.

A segment of a circle is the region between an arc and the chord of the circle. The chord connects the two endpoints of the arc.

Angles in the same segment of a circle are equal. This result follows from the property of inscribed angles that subtend the same arc. Equal arcs yield equal inscribed angles due to geometric congruence.

A semicircle subtends a right angle at any point on the circle. This follows from Theorem 9.8 and its special case where the arc is a semicircle and the central angle is 180°, so the inscribed angle becomes 90°.

If a line segment subtends equal angles at two points on the same side of the line then all four points lie on a circle. This is a condition for concyclicity proven geometrically.

If the sum of either pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. This is derived from the angle properties of cyclic segments and verified by construction.

Theorem proves that if opposite angles of a quadrilateral are supplementary then it is a cyclic quadrilateral. This is proven using angle properties of arcs and chords.

A circle divides the plane on which it lies into three parts: the interior, the exterior and the circle itself.

Equal chords of a circle subtend equal angles at the centre. This is because the angle subtended by a chord at the centre depends only on the chord’s length.

When a diameter is used to form a semicircle the angle subtended by this arc at any point on the remaining part of the circle is 90°.

The chords intersecting and making equal angles with a diameter from the point of intersection are equal in length. This is a direct consequence of triangle congruence and angle equality as discussed in the example of Theorem.

A diameter subtends a right angle at any point on the circle because it creates a semicircle. This result is a special application of Theorem and is often used in problems involving triangles inscribed in a semicircle.

When two chords intersect and subtend equal angles with a diameter through the point of intersection then the chords are equal. This follows from congruence of triangles formed by connecting the chords with the circle’s centre.

A cyclic parallelogram must have its opposite angles supplementary. Since opposite angles of a parallelogram are equal this implies each angle must be 90°, so the figure becomes a rectangle.

The perpendicular from the centre of a circle to a chord bisects the chord. This is a fundamental property of circles.

There is one and only one circle passing through three given non-collinear points. This is because three non-collinear points uniquely determine a circle.

The perpendicular from the centre to a chord bisects the chord. The triangle formed is isosceles and using congruence rules this midpoint property is established and verified through construction.

Equal arcs of a circle correspond to equal chords. This is a consequence of the congruence between arcs and the subtended chords. It follows from the definition of arc congruence.

The angle subtended at the centre by an arc is twice the angle subtended by the same arc at any point on the remaining circle. This is a key property used in solving circle geometry problems.

In a cyclic quadrilateral the sum of opposite angles is 180°. So, ∠A + ∠C = 180°. This is shown using the properties of arcs and the way they subtend angles at different vertices.

If a quadrilateral’s opposite angles are supplementary then all its vertices lie on a circle. This confirms it as a cyclic quadrilateral and is used in cyclic proofs.

Angles in the same segment of a circle are equal. This is a fundamental property of circles.

Using the Pythagorean theorem:
r² = d² + (chord/2)².
Substitute r = 10 cm, chord/2 = 6 cm:
10² = d² + 6²
→ 100 = d² + 36
→ d² = 64
→ d = 8 cm.