Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.
12. Cheatsheet
1) Area & Circumference
Circumference of a circle = 2πr Area of a circle = πr2
Circumference of an arc = 3600θ02πr
Area of a sector = 3600θ0πr2
2) Chord Properties
(a) A line drawn from the centre of a circle to a chord's midpoint is perpendicular to the chord & vice-versa.
(b) Equal chords subtend equal angles at the centre.
(c) Equal chords subtend equal angles on the circle along their respective major or minor arcs.
(d) Equal chords are equidistant from the centre of the circle & vice-versa.
(e) When two chords AB and CD intersect at O, then AO × BO = CO × DO.
3) Angle Properties
(a) All angles subtended by an arc on the same side of its segment are equal.
(b) Angles subtended by the end points of the diameter at any point on circle is 900.
(c) Angles subtended by two points of the circle at the centre is twice the angle subtended by those two points at any other point in the major segment, i.e., Central Angle = 2 × Inscribed Angle.
4) Cyclic Quadrilateral: A quadrilateral which can be circumscribed by a circle
(a) the sum of opposite angles of a cyclic quadrilateral are equal and add up to 1800.
(b) The exterior angle of a quadrilateral is equal to the interior opposite angle.
(c) Where AC and BD are the diagonals of Cyclic Quadrilateral ABCD, AC × BD = AB × CD + BC × DA
(d) Area = (s−a)(s−b)(s−c)(s−d)
5) Tangential Quadrilateral: A quadrilateral which can be inscribed by a circle, wherein the sides are the tangents of a circle. The sum of opposite sides of a tangential quadrilateral are equal.
6) Tangents:
(a) From any external point, exactly two tangents can be drawn to meet a circle, whose lengths are equal.
(b) From any point on a circle, only one tangent can be drawn.
(c) The line joining the centre to the tangential point is perpendicular to the tangent.
(d) If two tangents are drawn to a circle from an external point, then the line joining the the centre of the circle and this external point,
(i) bisects the angle subtended by the tangents at the external point
(ii) bisects the central angle formed by the points where the common tangents touch the circle
(iii) is perpendicular to the line joining the points where the common tangents touch the circle
(e) Where PT is the tangent and PB is the secant intersecting the circle at A and B, PT2 = PA × PB
(f) The angle subtended by a chord and a tangent on one side equals the angle subtended by the chord in the alternate segment.
7) Where AB and CD are the direct tangents, and EF and GH are the transverse tangents and where r1 & r2 are the radii of the circles with O & P as centres,
AB2 = CD2 = OP2 - (r1 - r2)2
EF2 = GH2 = OP2 – (r1 + r2)2
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