Alternate Segment Theorem

In geometry, the alternate segment theorem states that the angles created by a secant and a tangent, from the same point outside a circle, are equal. A secant is a line that intersects a circle in two points. A tangent is a line that intersects a circle in one point.

How the Theorem Works

The theorem works by drawning a secant, or a line that intersects the circle in two points, and then drawing the two tangents from those points of intersection. Because the tangents are perpendicular to the secants at those points of intersection, right triangles are formed. The theorem states that the angles marked "a" in the diagram below are equal.

More About the Theorem

The theorem is also sometimes called the inscribed angle theorem. It can be proven using basic algebraic manipulations. It is a fundamental theorem in geometry that has many applications in other disciplines such as physics and engineering.

The alternate segment theorem is a fundamental geometric principle with many applications. Understanding and being able to apply this theorem can give you a leg up infields such as physics and engineering that build on geometric principles.

FAQ

What is alternate segment theorem proof?

The alternate segment theorem is a statement in Euclidean geometry that states that the sum of the lengths of the segments between the points of intersection of a line with a conic section is constant. A proof of this theorem was first given by Pappus of Alexandria in his Mathematical Collection.

What is an alternating segment?

An alternating segment is a line segment that intersects a conic section in two points. The segments between the points of intersection are called the alternate segments of the conic section.

How do you identify an alternate segment?

An alternate segment can be identified by its length. The segments between the points of intersection of a line with a conic section are called the alternate segments of the conic section. The length of an alternate segment is the difference between the lengths of the two segments into which it divides the line.

What is the difference between an alternate segment and a secant segment?

A secant segment is a line segment that intersects a conic section in two points. An alternate segment is a line segment that intersects a conic section in two points and has the same length as the segments into which it divides the line.

What is the difference between an alternate segment and a tangent segment?

A tangent segment is a line segment that intersects a conic section in one point. An alternate segment is a line segment that intersects a conic section in two points and has the same length as the segments into which it divides the line.

What are the 8 circle theorems?

The circle theorem states that a circle is a locus of points that are equidistant from a fixed point, called the center. There are 8 circle theorems:

The eight circle theorems are:

1) The centre of a circle is equidistant from all points on the circumference.

2) A line segment joining two points on the circumference of a circle passes through the centre.

3) A line segment joining two points on the circumference of a circle is twice as long as the radius.

4) A line segment joining two points on the circumference of a circle is perpendicular to the radius at the midpoint.

5) A line segment joining two points on the circumference of a circle subtends equal angles at the center.

6) Angles in the same segment of a circle are equal.

7) The angle between a radius and a tangent drawn to a point on the circumference is equal to the angle subtended by the arc at the center.

8) The angle in a semicircle is a right angle.