Introduction on Circle

1. Circles and some related terms A circle is defined as a set of all such points in a given plane which lie at a fixed distance from a fixed point in the plane. This fixed point is called the  cente r  of the circle and the fixed distance is called the  radius  of the circle where point P is the center of the circle and segment PQ is known as the radius. The radius is the distance between all points on the circle and P. It follows that if a R exists such that ( seg.PQ ) > ( seg.PR) the R is inside the circle. On the other hand for a T if (seg.PT ) > ( seg.PQ ) T lies outside the circle, and if (seg.PS ) = ( seg.PQ ) it can be said that S lies on the circle.

2. Lines of a Circle The lines in the plane of the circle are classified into three categories. a) Lines like l which do not intersect the circle. b) Lines like m which intersect the circle at only one point. c) Lines like n which intersect the circle at two points..

3. Lines of a Circle Lines like m are called  tangents . A tangent is a line that has one of its points on a circle and the rest outside the circle . Thus K is the point of tangency. Line n is called a  secant  of the circle. A secant is defined as any line that intersects a circle in two distinct points. K

4. Lines of a Circle A segment whose end points lie on a circle is called a  Chord  . In a figure AB is a chord of the circle. Thus a chord is always a part of secant. Thus the other chords are;

5. Lines of a Circle The longest chord of the circle passes through its center and is called as the  diameter . In the figure chord CD is the diameter. It can be noticed immediately that the diameter is twice the radius of the circle. The center of the circle is the midpoint of the diameter.

6. Example 1 Refer to Name the center of Name the longest chord Name three radii Name a secant Name a tangent and the point of tangency. If OC=12, find SI. Is OS a chord of . Is SI ER? Explain.  

7. Arcs The angle described by any two radii of a circle is called the central angle. Its vertex is the center of the circle. APB is a central angle. The part of the circle that is cut by the arms of the central angle is called an arc. AB is an arc   And the other arcs are;

8. Arcs a rcAB is called the minor arc and  is the arcAOB is a major arc . The minor arc is always represented by using the two end points of the arc on the circle. However it is customary to denote the major arc using three points. The two end points of the major arc and a third point also on the arc.

9. Arcs If a circle is cut into two arcs such that there is no minor or major arc but both the arcs are equal then each arc is called a  semicircle .

10. Arcs An arc is measured as an angle in degrees and also in units of length. The measure of the angle of an arc is its central angle and the length of the arc is the length of the portion of the circumference that it describes. If is a central angle, then  

11. The Arc Addition Postulate Given point B on , t hen = +  

12. Example 2 Identify the following. 2 major arcs 2 minor arcs An acute central angle An obtuse central angle A radius which is not a part of a diameter A semicircle

13. Inscribed angles An inscribed angles are formed by chords. the vertex O of the inscribed AOB is on the circle. The minor arc AB cut on the circle by an inscribed angle is called as the intercepted arc.  

14. The Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. If is an inscribed angle, then  

15. Example 3 In the figure at the right, =72. Find  

16. Inscribed Angle Theorem:  If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure. If and are inscribed angles with same intercepted arc CD, then .  

17. Example 4 Given m and Find  

18. The Semicircle Theorem An angle inscribed in a semicircle is a right angle. If then  

19. Example 5 Solve the value of x.