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Geometry: Tangents and Secants





 



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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.

A tangent is a line that only intersects a geometric figure such as a curve at only one location. If a line intersects an object in geometry more than once then it is not a tangent.

For the complete circle, however, it is possible to say that a tangent is a line that intersects the circle at only one point.

A tangent for a circle only shares a single point with the circle.

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.

A secant is a line that intersects an object in geometry more that once.

The point of tangency is the point of intersection of a tangent line with the circle.

Common Tangent
A line that is tangent to more than one circle is a common tangent to these circles.


A common tangent intersects multiple geometric objects at only one place for each object.

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.


An external common tangent for two circles does not cross the line segment connecting their centers.

Common Internal Tangent
A common internal tangent intersects the line segment connecting the centers of the associated circles.


An internal common tangent for two circles does cross the line segment connecting their centers.

Tangent Circles

Tangent 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.


Internal tangent circles are coplanar circles that touch at only one point and are on the same side of the tangent line that passes through that point.
External Tangent Circles
If the tangent circles are on opposite sides of the tangent in the plane then they are external tangent circles.


External tangent circles are coplanar circles that touch at only one point and are on different sides of the tangent line that passes through that point.

Tangent Segments and Angles

Tangent Segment
A tangent segment is a segment of the tangent line with the point of tangency as one of its endpoints.


A tangent segment is part of a tangent line with one of its endpoints as the point of tangency.

Tangent Angle
A tangent angle is the angle formed by two intersecting tangents or tangent segments.


A tangent angle is formed by any combination of tangent lines or tangent segments.

Secant Segments and Angles

Secant Segment
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 segment is part of a secant line with one of its endpoints as a point of intersection with the geometric figure.

Secant Angle.
A secant angle is the angle between two secants or secant segments where the vertex does not lie on the circle.


A secant angle is formed by two secant lines or two secant segments with the intersection or vertex outside 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.


The tangent line of a circle is perpendicular to the radius of the circle.

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. The angle formed by a chord and a tangent is half the measure of the intercepted arc.

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.


A secant angle with its vertex inside the circle is half the sum of the measures of the intercepted arcs.

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.


A secant angle with its vertex outside the circle is half the difference of the measures of the intercepted arcs.

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.

The angle formed by the intersection of a secant and a tangent at the point of tangency is half the measure of the intercepted arc.

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.


The angle formed by the intersection of a secant and a tangent outside the circle is half the measure of the difference of the intercepted arcs.

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.


When two tangents intersect, the angle formed is half the measure of the difference between the intercepted arcs.

 




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