Points, Lines, Planes
Planes and Space
Tangents and Secants
Chords, Power Point
Congruent Circles, Arcs
Symmetry and Transformations
Rules Postulates Theorems
A tangent is usually though of as a line that touches a curve at only one point. This can be used in defining a tangent for a circle but not for other types of figures in geometry. Another way to think of a tangent is that it is a line whose slope is the same as the curve that it touches at a point on the curve.
For the complete circle, however, it is possible to say that a tangent is a line that intersects the circle at only one point.
There is another type of line that intersects a circle in two places. This line is called a secant. It can be defined as a line that intersects a circle at two points.
The point of tangency is the point of intersection of a tangent line with the circle.
A line that is tangent to more than one circle is a common tangent to these circles.
There are two kinds of common tangents depending on how they are oriented with the circles.
Common External Tangent
Common external tangents do not intersect with the line segment connecting with the centers of the circles associated with the tangents.
Common Internal Tangent
A common internal tangent intersects the line segment connecting the centers of the associated circles.
Two or more circles are tangent circles if they intersect the same tangent at the same point.
Internal Tangent Circles
Internal tangent circles are coplanar with their centers located on the same side of the tangent.
External Tangent Circles
If the tangent circles are on opposite sides of the tangent in the plane then they are external tangent circles.
Tangent Segments and Angles
A tangent segment is a segment of the tangent line with the point of tangency as one of its endpoints.
A tangent angle is the angle formed by two intersecting tangents or tangent segments.
Secant Segments and Angles
A secant segment is a line segment that intersects a circle at two points with exactly one point of this intersection as an endpoint.
A secant angle is the angle between two secants or secant segments where the vertex does not lie on the circle.
Theorems For Circle Tangents And Secants
Tangent Perpendicular to Radius
If a line or line segment is tangent to a circle then it is perpendicular to the radius of the circle at the point of tangency.
Line Perpendicular to Radius
If a line is perpendicular to the radius of a circle at the endpoint of the radius on the circle then the line is a tangent line.
Tangent To Chord Angle
The measure of an angle formed by a chord and a tangent is half the measure of the arc contained within that angle.
where O is the center of the circle.
Secant Angle Vertex Inside Circle
The measure of a secant angle with its vertex inside the circle is half the sum of the measures of the arcs contained by the angle itself and its vertical angle.
Secant Angle Vertex Outside Circle
The measure of a secant angle with its vertex outside the circle is the difference of the arcs intercepted by the secants.
Secant Tangent Intersection
With the vertex at the point of tangency, if a secant and a tangent intersect each other at the point of tangency of a circle then the measure of the angle formed is one-half the measure of the arc contained in the angle.
This geometry rule is similar to the one for the chord and tangent angle.
Exterior Secant Tangent Intersection
If a secant and a tangent intersect outside a circle, the angle formed by their intersection is equal to the difference of the intercepted arcs on the circle.
Intersection of Tangents
The angle formed by the intersection of two tangents is equal to the difference of the arcs contained within the angle on a circle.