4 Circle Std IX Subject Geometry Introduction Respected
4 Circle Std : IX Subject : Geometry Introduction : Respected sir, in previous standards we have already taught the techniques of drawing a circle of given radius , the distance between the centre of the circle and the chord , property of perpendicular drawn from the centre of the circle to its chord. , the property of distances between the centre and congruent chords of that circle. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
Comparison between previous units and upgraded units of circle Units- arc of a circle, types of arc, central angle, angular measure of arc, angle subtended by arc in previous syllabus are deleted and included in the upgraded syllabus of X th std. But theorem , there is one and only one circle passing through the three non collinear points is newly added. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
Subunits : 4. 1 Circle , related terms 4. 2 : points in a plane of circle 4. 3 : Circle passing through given points 4. 4 : Circles in a plane a) Concentric circles b) Congruent circles c) Intersecting circles 4. 5 Circle and line in a plane. 4. 6 : Angle subtended by chord ( Theorem and converse) ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
Defination of circle : A circle is a locus of points in a plane which are a constant distance from a fix point in that plane. Here fix point is called the centre of the circle. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
A r M. r B and constant distance is called the radius of the circle. The radius is shown by the line segment from the centre to the circumference. The circle consist of all those points whose distance from the centre of the circle equals the radius. Diameter of the circle d = 2 r ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
m B Now we can recall previous knowledge of students which are obtained from above activities. A. O D l C 4. 1 Here Point O is the centre of the circle. Seg OC seg AD line m seg. AB line l -------- ----- ‡µÖ¢ÖÖ 9 ----×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
. T Points in plane of the circle Circle is a simple closed A figure which divides the plane into three disjoint parts. Sr. no 1 2 3 Position In the interior In the exterior On the circle ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - B. M. O. P . L. Q . F Name of the. Dpoints ………………. . …………………… ¯ÖÏ� ú¸ü� Ö - Circle . G
Point O is the centre and r is the radius of the circle If • OA = r then A is on the circle • OA > r then A is in the exterior of the circle • OA< r then A is in the interior of the circle . T B A . O. P. F ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - . M . L. Q D ¯ÖÏ� ú¸ü� Ö - Circle . G
Circles in a plane: Given two circles in a Plane, there are following possibilities Concentric circles: Circles having same centre concentric circles and different radii are called Concentric circles. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - B O ¯ÖÏ� ú¸ü� Ö - Circle A
2) Congruent circles: Circles having equal radii are called congruent circles. Here PA and QB are circles with centers P and Q are congruent circles As PA = QB Q P A ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - B ¯ÖÏ� ú¸ü� Ö - Circle
Intersecting circles: l) coplanar circles having two points in common are called intersecting circles. Here circles with centers P and Q are intersecting each other in two points namely A and B. A. P . Q B ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
ii) Coplanar circles intersecting in one point are called touching circles. In fig 4. 15 and 4. 16 circles with centres P and Q are touching in point A. . Q. P ‡µÖ¢ÖÖ 9 A ×¾Ö ÖµÖ - . P A . Q ¯ÖÏ� ú¸ü� Ö - Circle
Circle and a line in the Plane Write relation between a circle and a line given in the In fig (a) , circle and line l are disjoint sets. In fig (b) line / is interesting the circle with centre O in only one Point, In fig (c) line l is intersecting the circle with centre O in two points A and B. In this case line I is called a "secant". O . Q . P B l ‡µÖ¢ÖÖ 9 A ×¾Ö ÖµÖ - P ¯ÖÏ� ú¸ü� Ö - Circle l
Secant: The line in the plane of the circle which intersects the circle in two distinct points is called a secant. Note: ln fig (b) seg AB is a chord. Every secant of a circle contains chord. : Diameter : Chord passing through the centre of the circle is called a diameter. Diameter is the largest chord of the circle. Every diameter is a chord, But Every chord is not a diameter. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
Angle Subtended by the chord Angle subtended by the chord at the centre. ÐAOB is the angle subtended by chord AB at the centre O. (i)Angle subtended by the chord at any point on the circle. ÐAPB is the angle subtended chord AB at point P. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - O A B P . O A ¯ÖÏ� ú¸ü� Ö - Circle B
¸ P Angle subtended by the chord any point inside the circle. ÐAPB is the angle subtended chord AB at point P in the interior of the circle. . O A B P (iii) Angle subtended by the chord any point outside the circle. ÐAPB is the angle subtended chord AB at point P in the exterior of the circle. . O A ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle B
Theorem 4. 2: The perpendicular from the centre of a circle to a chord bisects the chord. P A E B Converse 4. 3: The segment joining the mid point of a chord and the centre of a circle is perpendicular to the chord. P C ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle M D
Theorem 4. 4: The perpendicular from the centre of a circle to a chord bisects the chord. P A E B Theorem 4. 5 : The segment joining the mid point of a chord and the centre of a circle is perpendicular to the chord. P C ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle M D
Theorem 4. 4 : Congruent chords of congruent circle are equidistant from the centre. P A M Q B C N D Theorem 4. 5 Converse : If the angles subtended by chords at the centre of the circle are congruent , then the corresponding chords are equal. ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
Theorem : Chords which are equidistant from the centre of the congruent circle are congruent. P A ‡µÖ¢ÖÖ 9 M Q B ×¾Ö ÖµÖ - C N D ¯ÖÏ� ú¸ü� Ö - Circle
Theorem : If the angles subtended by chords at the centre of a circle are congruent then the corresponding chords are congruent. S R P O Q ‡µÖ¢ÖÖ 9 ×¾Ö ÖµÖ - ¯ÖÏ� ú¸ü� Ö - Circle
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