Geometry

Circles: Segments and Lengths

Big Picture

Along with angle measurements are the segments and lengths of a circle. Knowing the theorems can be useful in solving for chords in a circle.

Key Terms

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

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

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

Chords

Theorem: In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.

Excluded Values
  • GB = GE, so \(\overline{AC} \cong \overline{DF}\)

When two chords intersect in the interior of a circle, each chord is divided into two segments called segments of the chord.

Theorem: If two chords intersect a circle so that one chord is divided into segments of lengths a and b and the other chord into lengths c and d, the product of the segments of one chord is equal to the product of segments of the second chord.

 two chords intersect a circle
  • In the picture above, \(\color{blue}{ab} \color{black}{ = } \color{red}{cd}\).

Secants

A secant segment contains a chord of a circle and has exactly one endpoint outside the circle. The part of the secant segment outside the circle is called an external segment.

  • Segment a is a tangent segment.
  • Segment b is called an external segment.
  • Segment b + c is the secant segment.
Secants

Theorem: If two secants are drawn from a common point outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

  • Then a(a + b) = c(c + d)
external segment.

Geometry

Circles: Segments and Lengths Cont.

Tangents

Theorem: If two tangent segments are drawn from the same point outside the circle, then the segments are equal.

  • If \(\overline{CD}\) and \(\overline{CB}\) are tangent segments, then CD = CB
Tangents

Theorem: If a tangent and a secant are drawn from a common external point, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

  • Then \(a(a + b) = c^2\)
tangent segment.

Notes