Analytic Foundations of Geometry

Robert S. Wilson

5. Arc Length, Angles, and Rotations

Definitions

Let   A   and B be points on a circle. If the line that goes through   A   and   B   goes through the center of the circle, we say that   A   and   B   are diametically opposed.

If   A   and   B   are diametrically opposed, then the line through   A   and   B   divides the circle into two half circles.

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Let   A   and   B   be two points on a circle centered at   O.   The arc between   A   and   B   is the set of points on the circle which are in the angle between   OA   and   OB.

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If   A   and   B   are diametrically opposed, we will have to specify which half circle we mean.

Let   A   and   B   be two points on a circle. A partition of the arc between   A   and   B   is a set of points

 

A = P0, P1, P2, . . . , Pn = B

on the arc between   A   and   B   ordered so that   Pi   is in the angle between   OPi-1   and   OPi+1   where   O   is the center of the circle.

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The inside approximation of the arc length from   A   to   B   corresponding to the partition   A = P0 , P1, . . . , Pn = B   is the sum of the distances between consecutive points

|A, P1 | + | P1, P2 | + | P2, P3| + . . . + |Pn-1, B|

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Let   A   and   B   be two points on a circle. The outside approximation of the arc length from   A   to   B   corresponding to the partition   A = P0 , P1, . . . , Pn = B   is the sum of the distances starting at   A and going through all of the intersections of tangent lines at consecutive points.

|A, T1 | + | T1, T2 | + | T2, T3| + . . . + |Tn, B|

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Where   Ti   is the point where the tangent at   Pi-1   meets the tangent at   Pi.

One partition is called a refinement of another partition if every point in one partition is also a point in the other one.

Let   A   and   B   be two points on a circle. The length of the arc between   A   and   B   or the arc length between   A   and   B   is the least upper bound of all of the inside approximations.1

The unit circle is the circle of radius   1   centered at the origin.

In the angle between   AB   and   AC,   the ray from   A   thorough   B   and the ray from   A   through   C   are called the arms of the angle.

The angle illustrated above is called   / BAC.   The middle letter indicates the vertex of the angle, and the other two letters are points on the two arms of the angle.

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For any angle, the ratio of the arc length of the arc of a circle centered at the vertex of the angle bewtween the arms of the angle to the radius of the circle is called the number of radians in the angle, or the radian measure of the angle.2 The notation for the number of radians in an angle   / BAC   will be   m/ BAC.

To get the number of degrees in an angle, multiply the number of radians by   180o/pi3

A straight angle is one whose two rays are opposite rays of the same straight line.

If the arms of an angle are perpendicular, the angle is called a right angle.

The number of radians in a straight angle is denoted by pi. Pi is a Greek letter that looks like

A straight angle has   180   degrees.

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An angle which is smaller than a right angle is called an acute angle.

An angle which is between a right angle and a straight angle in size is called an obtuse angle.

If two angles add up to a straight angle, they are called supplementary.

Supplementary angles

If two angles add up to a right angle they are called complementary.

Complementary angles

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If two lines cross in a plane as in the figure below, the angles which are opposite each other called vertical angles.

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A rotation is accomplished by fixing one point of the plane and moving all the points through a fixed angle about the fixed point.

In the figure,   A   is moved to   A',   B   is moved to   B',   and   C   is moved to   C'.   In order for the movement to be a rotation we must have

|OA| = |OA'|

|OB| = |OB'|

|OC| = |OC'|

and   / AOA',   / BOB',   and   / COC'   all have the same size.

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If you start at a point on a circle where a horizontal line through the center meets the circle in the negative   x   direction and proceed in the positive  y   direction, you are going clockwise.

The opposite direction along a circle from clockwise is called counterclockwise.

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Analytic Foundations of Geometry

R. S. Wilson