A circle is the set of all points in a plane equidistant from a fixed point called the centre. The constant distance is the radius (r).
- Key Terms:
- Chord: A line segment joining two points on the circle. The diameter is the longest chord passing through the centre.
- Arc: A part of the circle. A minor arc is less than a semicircle; a major arc is more than a semicircle.
- Sector: Region bounded by two radii and an arc.
- Segment: Region bounded by a chord and an arc.
- Tangent: A line touching the circle at exactly one point; perpendicular to the radius at that point.
- Secant: A line intersecting a circle at two points.
- 1.Important Theorems:
- 2.Equal chords are equidistant from the centre; chords equidistant from the centre are equal.
- 3.The perpendicular from the centre to a chord bisects it (and conversely).
- 4.Angle subtended at centre = 2 x angle subtended at any point on the remaining arc.
- 5.Angles in the same segment are equal.
- 6.Angle in a semicircle = 90 degrees.
- 7.Opposite angles of a cyclic quadrilateral are supplementary (sum = 180 degrees).
Chord AB of a circle has its midpoint M. The line from centre O to M is:
By Theorem 2, OM is perpendicular to AB.
The angle subtended by arc PQ at the centre is 100 degrees. Find the angle at any point on the major arc.
Angle at centre = 2 x angle at circumference. Angle at circumference = 100/2 = 50 degrees.
In a cyclic quadrilateral ABCD, angle A = 80 degrees. Find angle C.
Opposite angles are supplementary: angle C = 180 - 80 = 100 degrees.
Angle in a semicircle. If AB is a diameter and C is on the circle, find angle ACB.
By theorem, angle ACB = 90 degrees.
Two chords PQ and RS of a circle are equal. If PQ is 3 cm from the centre, how far is RS?
Equal chords are equidistant from the centre, so RS is also 3 cm from the centre.
A chord of length 16 cm is at 6 cm from the centre. Find the radius.
Half-chord = 8. By Pythagoras: r2 = 82 + 62 = 64 + 36 = 100. r = 10 cm.
Angles subtended by a chord at two points on the same side of the chord: if angle APB = 70 degrees, find angle AQB where Q is on the same arc.
Angles in the same segment are equal. Angle AQB = 70 degrees.
Common mistakes
The theorem "angle at centre = 2 x angle at circumference" applies only when both angles are subtended by the same arc. Students also confuse the angle in a semicircle (always 90 degrees) with general angles in a segment.
Summary
Circle theorems connect the centre angle, inscribed angles, chords and cyclic quadrilaterals. The perpendicular from centre bisects chords. Opposite angles of a cyclic quadrilateral are supplementary. These theorems are frequently used in coordinate and proof-based problems.