Circles — Telangana (SSC) Class 9 Mathematics Solutions (Free)

A circle is the set of all points in a plane that are the same fixed distance (the radius) from a single fixed point (the centre). Three basic properties: 1. All radii are equal — since every point on the circle is, by definition, the same distance from the centre. 2. A chord is a line segment connecting any two points that both lie on the circle itself. 3. The diameter (a chord passing through the centre) is the longest possible chord — any chord not passing through the centre is necessarily shorter, since the direct path through the centre maximises the distance between two circle points. Given radius = 7 cm: Diameter = 2 × radius = 2 × 7 = 14 cm Circumference = 2πr = 2 × (22/7) × 7 = 44 cm

Theorem: the angle subtended by an arc at the centre of a circle is always TWICE the angle subtended by the same arc at any point on the remaining (major) part of the circle's circumference. Given: arc AB subtends 70° at the centre O. Angle at C (on the major arc) = ½ × angle at centre = ½ × 70° = 35° The arc subtends an angle of 35° at point C.

The perpendicular from a circle's centre to any chord always bisects that chord, and forms a right-angled triangle with the radius as the hypotenuse. Given: radius = 13 cm, perpendicular distance = 5 cm. Using the Pythagorean theorem: (half-chord)² = radius² − distance² (half-chord)² = 13² − 5² = 169 − 25 = 144 half-chord = √144 = 12 cm Full chord length = 2 × 12 = 24 cm

Theorem (Angles in the Same Segment): angles subtended by the same chord at any two different points lying on the same arc (segment) of a circle are always equal to each other. Given: chord AB subtends 55° at point C on the major arc. Since point D also lies on the SAME major arc (the same segment) as C, the angle AB subtends at D must equal the angle it subtends at C, by this theorem. The angle at D is 55°.

This uses the Intersecting Chords Theorem: when two chords intersect inside a circle, the products of the lengths of their respective segments are equal. AP × PB = CP × PD Given: AP = 8 cm, PB = 6 cm, CP = 4 cm. Find PD. 8 × 6 = 4 × PD 48 = 4 × PD PD = 48/4 = 12 cm

A chord divides a circle into two arcs — a major arc and a minor arc. A point on the major arc and a point on the minor arc, together with the chord's two endpoints, form a cyclic quadrilateral, whose opposite angles are supplementary (sum to 180°) — this is the Cyclic Quadrilateral theorem. Given: the chord subtends 70° at a point on the major arc. Since the angle at a point on the minor arc is the OPPOSITE angle in this cyclic quadrilateral to the angle at a point on the major arc: Angle at minor arc = 180° − 70° = 110° The chord subtends an angle of 110° at a point on the minor arc — confirming that a chord subtends a LARGER angle from the minor arc side than from the major arc side, since the minor-arc viewpoint is "closer" to the chord.

Given: radius = 21 cm, central angle = 60°. Arc length is the fraction of the full circumference corresponding to the given angle out of the full 360°: Arc length = (θ/360°) × 2πr Arc length = (60/360) × 2 × (22/7) × 21 Arc length = (1/6) × 132 Arc length = 22 cm Sector area is the corresponding fraction of the full circle's area: Sector area = (θ/360°) × πr² Sector area = (60/360) × (22/7) × 21 × 21 Sector area = (1/6) × 1,386 Sector area = 231 cm²

In a cyclic quadrilateral (one whose four vertices all lie on a single circle), opposite angles are always supplementary — they add up to 180°. Given: ∠A = 100°, ∠C = 80°, ∠B = 95°. Check the first pair: ∠A + ∠C = 100° + 80° = 180° ✓ Find ∠D using the second pair: ∠B + ∠D = 180° 95° + ∠D = 180° ∠D = 85° Verification of all four angles together: ∠A + ∠B + ∠C + ∠D = 100° + 95° + 80° + 85° = 360° ✓ (matching the general quadrilateral angle-sum property, as it must, while the cyclic-specific opposite-angle pairs 100°+80° and 95°+85° each separately confirm the 180° supplementary rule).