Theorem 12-15
Lengths of Intersecting Chords or Secants
Include Formula
Intersecting Chords or Secants (On the interior):
Intersecting Secants (On the exterior):
Intersecting Tangent and Secant (on the exterior):
Intersecting Chords or Secants (On the interior):
If two chords or secants intersect inside a circle, then the product of the lengths of the segments of one chord or secant is equal to the product of the lengths of the segments of the other chord or secant.
Formula: AB × BC = CD × DE, where AB, BC, CD, and DE are the segments of the chords or secants intersecting inside the circle.
Intersecting Secants (On the exterior):
If two secants intersect outside a circle, then the product of the length of a secant and its external segment is equal to the product of the length of the other secant and its external segment.
Formula: AD × AE = BD × BE, where AD and BD are the secant lengths and AE and BE are their external segments.
Intersecting Tangent and Secant (on the exterior):
If a tangent and a secant intersect outside a circle, then the square of the length of the tangent is equal to the product of the lengths of the secant and its external segment.
Formula: AT2 = BS × BE, where AT is the length of the tangent, BS is the length of the secant, and BE is the external segment of the secant.
Theorem 12-15 states the relationships between the lengths of intersecting chords or secants in a circle.
1. Intersecting Chords (On the interior):
In a circle, if two chords intersect inside the circle, the product of the lengths of the chord segments is equal. This can be represented by the formula:
AB * BC = CD * DE
2. Intersecting Secants (On the exterior):
In a circle, if two secants intersect outside the circle, the product of the lengths of the secant segments is equal. This can be represented by the formula:
AE * ED = BE * EC
3. Intersecting Tangent and Secant (on the exterior):
In a circle, if a tangent and a secant intersect outside the circle, the square of the length of the tangent segment is equal to the product of the lengths of the secant segments. This can be represented by the formula:
AC^2 = AE * EC