Circles

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.

A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.

• Chord: a line segment whose endpoints lie on the circle.
• Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle.
• Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
• Circumference: the length of one circuit along the circle itself.
• Tangent: a straight line that touches the circle at a single point.
• Secant: an extended chord, a straight line cutting the circle at two points.
• Arc: any connected part of the circle's circumference.
• Sector: a region bounded by two radii and an arc lying between the radii.
• Segment: a region bounded by a chord and an arc lying between the chord's endpoints.

                 

Properties

• The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
• The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry groupis the orthogonal group O(2,R). The group of rotations alone is the circle group T.
• All circles are similar.
  • A circle's circumference and radius are proportional.
  • The area enclosed and the square of its radius are proportional.
    • The constants of proportionality are 2π and π, respectively.
• The circle which is centred at the origin with radius 1 is called the unit circle.
  • Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
• Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Section 9.1: Chords and Arcs.

Objectives
1. Define a circle and its associated parts, and use them in constructions.
2. Define and use the degree measure of arcs.
3. Define and use the length measure of arcs.
4. Prove a theorem about chords and their intercepted arcs.

Definitions
A circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.
A radius (plural, radii) is a segment from the center of the circle to a point on the circle.
A chord is a segment whose endpoints line on a circle.
A diameter is a chord that contains the center of a circle.
See diagram 9-1 for visual assistance.

9-1


Definition: Central Angle and Intercepted Arc.
A central angle of a circle is an angle in the plane of a circle whose vertex is the center of the circle. An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the intercepted arc of the central angle.
9-2


Definition: Degree Measure of Arcs.
The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180°.

Example: The measure of a circle's central angle is 65°, find the measure of the circle's major arc.
Answer: This may seem complicated, but if you think about it, it really isn't. Central angle=Minor arc, central angle=65°, so that means the minor arc must be 65°. The degree measure for a full circle is 360°, so to find the major arc it's 360-65, which is 295°, the answer.

Arc Length.
If r is the radius of a circle and M is the degree measure of an arc of the circle, then the length, L, of the arc is given by the following: L=M/360°(2(pi)r).

Chords and Arcs Theorem.
In a circle, or in congruent circles, the arcs of congruent chords are equal.

Converse of the Chords and Arcs Theorem.
In a circle, or in congruent circles, the chords of congruent arcs are equal.

Section 9.2: Tangents to Circles.

Objectives:
1. Define tangents and secants of circles.
2. Understand the relationship between tangents and certain radii of circles.
3. Understand the geometry of a radius perpendicular to a chord of a circle.

Definitions: Secants and Tangents.
Secant: A line that intersects the circle at two points.
Tangent: A line in the plane of the circle that intersects the circle at exactly 1 point. The 1 point is known as point of tangency.

9-3


Tangent Theorem:
If a line is tangent to a circle, then the line is perpendicular to a radius of the circle drawn to the point of tangency.

Radius and Chord Theorem:
A radius that is perpendicular to a chord of a circle cuts the chord in half.

Converse of the Tangent Theorem:
If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is perpendicular to the circle.

Theorem:
The perpendicular bisector of a chord passes through the center of the circle.

Section 9.3: Inscribed Angles and Arcs.

Objectives:
1. Define inscribed angle and intercepted arc.
2. Develop and use the Inscribed Angle Theorem and its corollaries.

Definition: Inscribed Angle.
An angle whose vertex lies on a circle and whose sides are chords of the circle

Definition: Inscribed Angle Theorem.
The measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.

Example: I'm just coming up with this on the top of my head, but, say you want to make the angle measurements on a sports game, (it is a real job, trust me on that.) In a baseball game, the whole field is seen in a circle instead of the diamond were used to in baseball, a bat hits a ball at a certain angle, the angle is determined on where the ball hits the bat, and where the bat is when you make contact. Where the ball makes contact it makes a line going both ways and makes an arc intercepting the circle twice. If the inscribed angle is between 45° and 90° it's fair, if not it's foul. I truly hope that makes sense, it was really hard to explain.

Definitions: Right Angle and Arc-Intercept Corollaries.
Right Angle Corollary: If an angle intercepts a semicircle, then the angle is a right angle.
Arc-Intercept Corollary: If two inscribed angles intercept the same arc, then they have the same measure.