Introduction on Circle

Circles and some related termsA 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 center 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.

Lines of a CircleThe 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..

Lines of a CircleLines 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

Lines of a CircleA 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;

Lines of a CircleThe 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.

Example 1Refer to

1. Name the center of 2. Name the longest

chord3. Name three radii4. Name a secant5. Name a tangent and

the point of tangency.6. If OC=12, find SI.7. Is OS a chord of .8. Is SIER? Explain.

ArcsThe 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;

ArcsarcAB 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.

ArcsIf 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.

ArcsAn 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

The Arc Addition Postulate

Given point B on , then = +

Example 2Identify the following.

1. 2 major arcs2. 2 minor arcs3. An acute central angle4. An obtuse central

angle5. A radius which is not a

part of a diameter6. A semicircle

Inscribed anglesAn 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.

The Inscribed Angle TheoremThe measure of an inscribed angle is half the measure of its intercepted arc.

If is an inscribed angle, then

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

Inscribed AngleTheorem:

 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 .

Example 4Given m

and

Find

The Semicircle Theorem

An angle inscribed in a semicircle is a right angle.

If then

Example 5Solve the value of x.