Dynamics Of The Circle (1963)

[Music] a a [Music] in our visual World circles are everywhere whole circles parts of circles large circles what is a circle a circle is a set of points a const distance from a given Center it is named by stating its Center this is circle o let us review some of the most important terms related to it a line joining to any point on the circle is a radius congruent circles are defined as those that have equal radi a segment of a line joining two points on the circle is called a chord and a chord that passes through the center of the circle is a diameter between two given points on a circle there are two subsets of points these are called arcs the smaller Arc is called the minor Arc the larger Arc is called the major Ark when the given points and the center of the circle lie on a straight line the two arcs formed are called semicircles an angle formed by two radi is called a central angle we say that a central angle and a cord both intercept an arc the intercepted Arc is related to the central angle in the following way as point a moves along the circle to various positions radius OA of course follows it if the radius moves through all of 360° the point has moved around the entire circle if the radius moves through 1 half of 360° the point has moved halfway around the circle if the radius moves through 1/4 of 360° the point has moved 1/4 the way around the circle if radius OA rotates through any fraction of 360° point a moves along that same fractional part of the circle thus we may say that a central angle and its Arc have the same measure this principle is the basis of the compass rows a device used by Navigators to indicate Direction on it 0° points towards north 90° East 180° South and 270° West the intermediate directions fall between these Southwest for instance is 225° if in the same Circle we have two equal central angles we can prove that the two arcs that intercept are also equal in degree measure let us say that angle a o is 40° since a central angle and its Arc have the same measure Arc AB is also 40° the same is true of angle C OD and Arc CD the two arcs are therefore equal and can be made to coincide however this this principle does not apply to Circles of different sizes although a central angle of these concentric circles is the same it is easy to see that the arcs intercepted by it are not equal in length the complete statement then should be if two central angles of the same Circle or of congruent circles are equal their intercepted arcs are equal in degree and length as a second method of proving arcs equal let us consider their relationship with the chords that intercept them if triangle OAB is rotated to a new position we obtain another triangle congruent to it and if the triangles are congruent to each other then the chords ab and a prime B Prime are equal and the central angles a o and a prime o Prime are equal therefore whenever the chords are equal their intercepted arcs must be equal and conversely if the arcs ab and a prime B Prime are equal then it can be shown that their chords ab and a prime B Prime are equal other methods of proving arcs equal are similar to methods proving straight line segments equal just as equal line segments have the same linear measure so do the equal arcs of a circle have the same measure in degrees and just as we refer to the midpoints of a line segment we may refer to the midpoint of an arc we may also prove arcs equal by using the same postulates as are used to prove line segments equal for example the addition postulate if equals are added to equals the sums are equal if AB b equals CD then by adding BC to each we can see that AC equals BD similarly if Arc AB equals CD then by adding Ark B to each we can see that Ark AC equals BD to summarize we can prove arcs equal by proving that their central angles are equal proving that their chords are equal or by applying postulates in a circle if a series of chords are perpendicular to a given chord the only one to bisected is a cord that passes through the center of the circle in other words a diameter we can then say that if a diameter is perpendicular to a chord then it bcts the chord incidentally it also bisects an arc intercepted by the cord we can illustrate this from another point of view as you can see the only cord to pass through the midpoint of this cord and the center of the circle is the one that is perpendicular to the given cord we can then say that the perpendicular bis sector of a cord passes through the center of the circle and is a diameter let us now consider the distance from a cord to the center of of the circle the distance is the same as the measure of the perpendicular from the cord to the center as the cord moves closer to the center it grows larger the largest possible cord is a diameter at that point the distance from the center is zero then as it's distance from the center increases the cord becomes smaller thus we have shown experimentally that if two chords are unequally distant from the center the chord nearer to the center is the larger and the converse is true we can also will see that if two chords are equal they are equally distant from the center and conversely if two chords are equally distant from the center then they are equal let's review some of the concepts we have just considered if two central angles of the same Circle or of congruent cires CES are equal their intercepted arcs are equal and the converse is also true if two chords of a circle are equal their intercepted arcs must be equal and the converse is true if a diameter is perpendicular to a cord then it bisects the cord and its arcs we can also say that the perpendicular bis sector of a cord passes through the center of the circle if two chords are unequal the chord nearer to the center is the larger and the converse is true if two chords are equal they are equally distant from the center and the converse is also true a central angle and its Arc have the same measure all that from the study of the Dynamics of the circle e [Music]