Page 1 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. Page 2 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 The word ‘tangent’ touch and was introduced 1583. In the above figure, A' B' is a tangent to the circle The tangent to a circle two end points of its Important Theorems: Theorem 1: The tangent at any point of a circle is perpendicular to the through the point of contact. circle is a line that intersects the circle at ‘tangent’ comes from the Latin word ‘tangere’, introduced by the Danish Mathematician Thomas , there is only one tangent at a point of the circle. is a tangent to the circle. circle is a special case of the secant, when its corresponding chord coincide. Important Theorems: The tangent at any point of a circle is perpendicular to the through the point of contact. at only one point. ‘tangere’, which means to Thomas Fineke in only one tangent at a point of the circle. when the The tangent at any point of a circle is perpendicular to the radius Page 3 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 The word ‘tangent’ touch and was introduced 1583. In the above figure, A' B' is a tangent to the circle The tangent to a circle two end points of its Important Theorems: Theorem 1: The tangent at any point of a circle is perpendicular to the through the point of contact. circle is a line that intersects the circle at ‘tangent’ comes from the Latin word ‘tangere’, introduced by the Danish Mathematician Thomas , there is only one tangent at a point of the circle. is a tangent to the circle. circle is a special case of the secant, when its corresponding chord coincide. Important Theorems: The tangent at any point of a circle is perpendicular to the through the point of contact. at only one point. ‘tangere’, which means to Thomas Fineke in only one tangent at a point of the circle. when the The tangent at any point of a circle is perpendicular to the radius Remarks: 1. By theorem mentioned 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 Case 1: There is no tangent to a circle passing circle. Case 2: There is one and only one tangent to a lying on the circle Case 3: There are exactly two outside the circle mentioned above, we can also conclude that at any point on a circle there can be one and only one tangent. The line containing the radius through the point of contact is also sometimes ’ to the circle at the point. Number of Tangents from a Point on a Circle: There is no tangent to a circle passing through a point lying inside the There is one and only one tangent to a circle passing through a point lying on the circle There are exactly two tangents to a circle through a point lying above, we can also conclude that at any point on a circle The line containing the radius through the point of contact is also sometimes through a point lying inside the circle passing through a point circle through a point lying Page 4 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 The word ‘tangent’ touch and was introduced 1583. In the above figure, A' B' is a tangent to the circle The tangent to a circle two end points of its Important Theorems: Theorem 1: The tangent at any point of a circle is perpendicular to the through the point of contact. circle is a line that intersects the circle at ‘tangent’ comes from the Latin word ‘tangere’, introduced by the Danish Mathematician Thomas , there is only one tangent at a point of the circle. is a tangent to the circle. circle is a special case of the secant, when its corresponding chord coincide. Important Theorems: The tangent at any point of a circle is perpendicular to the through the point of contact. at only one point. ‘tangere’, which means to Thomas Fineke in only one tangent at a point of the circle. when the The tangent at any point of a circle is perpendicular to the radius Remarks: 1. By theorem mentioned 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 Case 1: There is no tangent to a circle passing circle. Case 2: There is one and only one tangent to a lying on the circle Case 3: There are exactly two outside the circle mentioned above, we can also conclude that at any point on a circle there can be one and only one tangent. The line containing the radius through the point of contact is also sometimes ’ to the circle at the point. Number of Tangents from a Point on a Circle: There is no tangent to a circle passing through a point lying inside the There is one and only one tangent to a circle passing through a point lying on the circle There are exactly two tangents to a circle through a point lying above, we can also conclude that at any point on a circle The line containing the radius through the point of contact is also sometimes through a point lying inside the circle passing through a point circle through a point lying 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 to the circle. Theorem 2: The lengths of tangents drawn from an external point to a circle are equal. From the above diagram, 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 Which gives us PQ = PR Remarks: 1. The theorem can also be proved by using the Pythagoras Theorem as follows: PQ 2 = OP 2 – OQ 2 = OP 2. Note that ? OPQ = ? QPR, i.e. the centre 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 The lengths of tangents drawn from an external point to a circle are From the above diagram, i.e. in right triangles OQP and ORP, (Radius of the same circle with centre O) (Common side for both the triangles) By RHS congruence condition, OQP ? ?ORP Which gives us PQ = PR (Corresponding side of congruent triangles) The theorem can also be proved by using the Pythagoras Theorem as follows: = OP 2 – OR 2 = PR 2 (As OQ = OR) which gives PQ = PR. OPQ = ? OPR. Therefore, OP is the angle bisector of centre lies on the bisector of the angle between the two tangents. The length of the segment of the tangent from the external point P and the length of the tangent from the point P The lengths of tangents drawn from an external point to a circle are n right triangles OQP and ORP, (Corresponding side of congruent triangles) The theorem can also be proved by using the Pythagoras Theorem as follows: (As OQ = OR) which gives PQ = PR. OPR. Therefore, OP is the angle bisector of lies on the bisector of the angle between the two tangents.Read More

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