Geometry

Properties of a Circle

Big Picture

The many parts of a circle include arcs, tangents, segments, and chords. All these parts have various properties and theorems associated with them.

Key Terms

Circle: The set of all points in a plane that are equidistant (same distance away) from a specific point called the center.

Radius: (radii, plural) A line segment from the center to any point on the circle.

Diameter: A line segment from one point on the circle to another that contains the center of the circle.

Tangent: A line, line segment, or ray that intersects a circle at exactly one point.

Point of Tangency: The point where the tangent touches the circle.

Chord: A line segment whose endpoints are on a circle.

Secant: A line that intersects a circle in two points.

Arc: A section of a circle.

Semicircle: An arc that measures 180°.

Major Arc: An arc larger than a semicircle.

Minor Arc: An arc smaller than a semicircle.

Central Angle: An angle formed between two radii of a circle with its vertex at the center.

Basic Properties of Circles

The center of a circle is a point, so the center is usually labeled with a capital letter like a point. The circle below is circle A, labeled \(\odot\)A.

Although the position of the center and the length of the radius may differ, all circles are similar to each other.

  • Two circles with the same radius but different centers are congruent circles.
  • Two circles with the same center but different radii are concentric circles.

For two coplanar circles (two circles lying in the same plane), the circles can intersect in two points, one point, or no points.

Basic Properties of Circles
all circles

Tangent circles are two circles that intersect in one point.

  • The tangent line drawn through the point of tangency can be either external or internal.
 tangency

Tangents to a Circle

Circles that are not tangent can share a tangent line called a common tangent. The common tangent can also be internally or externally tangent.

Tangents to a Circle

Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

  • Line \(\overleftrightarrow{CB}\) is tangent to \(\odot A\) if and only if \(\overleftrightarrow{CB} \perp \overline{AB}\)
Tangents to a Circle

Geometry

Properties of a Circle Cont.

Arcs and Central Angles

Arcs are labeled with \(\frown\) The letters used to label an arc are the points on the circle

There are three types of arcs:

  • Semicircle: An arc that measures 180°. “Semi” means half, so a semicircle is half of a circle. m\(\overset{ \huge\frown}{EHG}\) = m\(\overset{ \huge\frown}{EJG}\) = 180°
  • Minor arc: An arc that measures less than \(180°. m\overset{ \huge\frown}{BC} < 180°\)
  • Major arc: An arc that is greater than 180°. Use 3 letters to label a major arc to distinguish it from a minor arc! \(m\overset{ \huge\frown}{BDC} > 180°\)

The central angle divides a circle into two arcs: either two semicircles or one major arc and one minor arc.

Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of measures of the two arcs

  • \(m\overset{ \huge\frown}{CDB}\) = \(m\overset{ \huge\frown}{CD}\) + \(m\overset{ \huge\frown}{DB}\)
Arcs and Central Angles

Chords

The endpoints of a chord lie on the circle, so a chord divides into two arcs.

  • A diameter divides a circle into two semicircles.
  • All other chords divide a circle into a minor arc and a major arc.
Chords

Graphing a Circle

Standard equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\)

  • Center of circle = (h, k); radius = r

Since it’s (x - h) in the equation, remember to take the opposite sign of the value in the equation when finding h and k



To graph a circle:

  • First draw the center of the circle on the coordinate plane.
  • Then draw a circle around it with radius r.

To write the equation for a circle:

  • Plug the information given in the problem into the standard equation of a circle.
  • If we’re given some points that are on the circle, substitute their x- and y-coordinates for x and y in the equation to find h, k, and r.
Graphing a Circle

Notes