Geometry: Circles- arcs, chords, tangents, sector, segment, secant
This is a free lesson from our course in Geometry
 
   
In this lesson you’ll learn the basics and more details about some other parts of the circles that lie on or inside the circles but with special names, say an arc- major and minor, chords, tangents, sector, segment, and secant. It will help to understand deeper the relationship between- diameter and a chord when they are perpendicular to each other, the measures of an angle and its minor/ major arc, and measurement of an angle by two intersecting chords etc. In turn you’ll be able to solve the real world applications using the learning and developed skills. Included also are the relevant problems to support explanation. The contents will be presented by the instructor in own handwriting with the help of several examples with solutions and using watch video. (More text below video...)
<h2> Geometry - Circles- arcs, chords, tangents, sector, segment, secant</h2> <p> arc, circle, basics, terms, points, video, secant, segment, tangent, major arc, minor arc, diameter, geometry, chord, semi circle, sector, solution, minor segment, major segment, example, circumference, intercepted arc, geometry help, practice questions, quizzes</p> <p> An arc is a part of the circumference of a circle. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is measured in degrees and length.</p>
Other useful lessons:
Measuring Circles
Equation of a circle in standard form
Inscribed and Circumscribed Polygons
(Continued from above) Arcs:
An arc is a part of the circumference of a circle and defined as: 'an arc of a circle is the part of the circle between two points on the circle'. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is measured in degrees and length. If the measure of minor arc is i.e. the measure of the central angle intercepted by the minor arc, then the measure of major arc is (360 -) i.e. the measure of the central angle intercepted by the minor arc, then the measure of major arc is (360 - ). E.g. if measure of a minor arc is 100, then major arc is (360 - 100) = 260.
Now look at the relationship of radius and measure of central angle: if r is the radius of the circle and is the measure of central angle, then length of the arc intercepted by the angle is the measure of central angle, then length of the arc intercepted by the angle is given by  / 360 * 2r. E.g. if  = 120, then length of the arc intercepted is 4/3 units. Notice that an arc is measured commonly not by its length, but more often expressed as measure of the angle whose vertex is the center of the circle and its rays intercept the endpoints of the arc. Hence an arc can be anywhere from 0 to 360.
Types of Arcs:
major arc, minor arc, and semicircle
• If mAOB < 180, points A and B and the points of circle in the interior of AOB make up minor arc AB, written as
• Points A and B and the points of the circle not in the interior of AOB make up major arc AB, written as or
• If mAOC = 180, points A and C separate circle O into two equal parts, each of which is called a semicircle. In (Fig: 1), and two different semicircles
Major arc, Minor arc, and Semicircle
Congruent Arcs:
Central angle is an angle whose vertex is the center of a circle ‘O’. Any central angle intercepts the circle at two points, thus defines the arc (Fig: 2).
Central angle Congruent-Arcs
Congruent arcs are arcs of the same circle or of congruent circles that are equal in measure. In (Fig: 3, above), if O O' and mCD = mC'D' = 60, then
Arc Addition Postulate
If AB and BC are two arcs of the same circle having a common endpoint and no other points in common, then AB + BC = and mAB + mBC = m.
Theorems
• In a circle or in congruent circles, if central angles are congruent, then their intercepted arcs are congruent (Figure below).
if O O', AOB COD, and AOB A'O'B'.
then and
Arc Addition Postulate
• In a circle or in congruent circles, central angles are congruent if their intercepted arcs are congruent.
• In a circle or in congruent circles, central angles are congruent if and only if their intercepted arcs are congruent.
E.g. (In figure below) P, Q, S, and R are points on circle O, mPOQ = 100,mQOS = 110, and mSOR = 35. Find  mROP
Arc Addition Postulate
mROP will be- [360 – (100 + 110 + 35)] = 115, as the final answer.
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