CIRCLES Circle: A circle is a collection of all points in a plane which are at a constant distance from a fixed point. Some parts of a circle: Chord: A line segment joining any two end points on the circle is called a chord. Diameter is the longest chord. Secant: A line which intersects the circle in two distinct points is called secant of the circle. Tangent: A line which touches the circle at only one point is called tangent to the circle. Three different situations that can arise when a circle and a line are given in a plane, consider a circle and a line PQ. There can be three possibilities as given: (i) The line PQ and the circle have no common point. Here, PQ is called a non-intersecting line with respect to the circle. (ii) There are two common points A and B that the line PQ and the circle have. Here, we call the line PQ a secant of the circle. (iii) There is only one point A which is common to the line PQ and the circle. Here, the line PQ is called a tangent to the circle. Tangent to the circle: A tangent to a circle is a line that intersects the circle at only one point. The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish Mathematician Thomas Fineke in 1583. In the above figure,, there is only one tangent at a point of the circle. A' B' is a tangent to the circle. circle The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide. Important Theorems: Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Remarks: 1. By theorem mentioned above, we can also conclude that at any point on a circle there can be one and only one tangent. 2. The line containing the radius through the point of contact is also sometimes called the ‘normal’’ to the circle at the point. Number of Tangents from a Point on a Circle Circle: Case 1: There is no tangent to a circle passing through a point lying inside the circle. Case 2: There is one and only one tangent to a circle passing through a point lying on the circle Case 3: There are exactly two tangents to a circle through a point lying outside the circle Note: The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent from the point P to the circle. Theorem 2: The lengths of tangents drawn from an external point to a circle are equal. From the above diagram, i.e. inn right triangles OQP and ORP, OQ = OR (Radius of the same circle with centre O) OP = OP (Common side for both the triangles) Also, ∠OQP = ∠ORP ⇒ By RHS congruence condition, ∆OQP OQP ≅ ∆ORP Which gives us PQ = PR (Corresponding side of congruent triangles) Remarks: 1. The theorem can also be proved by using the Pythagoras Theorem as follows: PQ2 = OP2 – OQ2 = OP2 – OR2 = PR2 (As OQ = OR) which gives PQ = PR. 2. Note that ∠OPQ = ∠OPR. Therefore, OP is the angle bisector of ∠QPR, i.e. the centre lies on the bisector of the angle between the two tangents.

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