Theorems on Segments

What is Theorem:

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

What is Theorem in Geometry:

A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven. Listed below are six postulates and the theorems that can be proven from these postulates. Postulate 1: A line contains at least two points.

Segment:

Alternate segment theorem. The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment. This is the circle property that is the most difficult to spot. Segments of Tangents and Secants. In the figure, is called a tangent secant because it is tangent to the circle at an endpoint. Similarly, is a secant segment and is the external segment of . The external segments are those that lie outside the circle.

Angles in the same segment are equal. Angles subtended (made) by the same arc at the circumference are equal.

Segments from Secants

When two secants intersect outside a circle, the circle divides the secants into segments that are proportional with each other.

Two Secants Segments Theorem: If two secants are drawn from a common point outside a circle and the segments are labeled as below, then a(a+b)=c(c+d).

Example 1

Find x. Simplify any radicals.

Use the Two Secants Segments Theorem.

8 ( 8 + x ) = 6 ( 6 + 18 )

64 + 8x = 36 + 108 = 144

8x = 144 – 64 = 80

8x = 80

x = 10