Measure of Arcs and Angles Formed by Intersecting Secants and Tangents. Inside and Outside the Circle.

Should you be given a problem with a secant and a tangent intersecting in the exterior of the circle; would you be able to find the measure of one of the intersected arcs, if you are given the other intersected arc and the angle at the point of intersection? What if they ask you to find the angle when the two referred arcs are given? If they give you two intersecting tangents ¿would you be able to do the same?

In this lesson we strife to give you a detailed presentation of the involved theorems when you have a tangent and a secant intersecting at the point of tangency, a tangent and a secant intersecting at an exterior point, and two tangents, or two secants intersecting at an exterior point; followed by a step by step solution of several examples. We are sure your knowledge will not escape through the tangent!

Lesson's Content

Lesson's Glossary

Arc
The curved segment that is between two points in the circumference of a circle.

Arc length
The distance between an arc's endpoints along the path of the circle.

Center of a circle
The point that all points in the circle are equidistant from.

Central angle of a circle
An angle whose vertex is the center of the circle.

Chord of a circle
A segment whose endpoints are on a circle.

Circle
The set of points on a plane at a certain distance (radius) from a certain point (center); a polygon with infinite sides.

Minor arc
An arc whose endpoints form an angle less than 180 degrees with the center of the circle.

Major arc
An arc whose endpoints form an angle over 180 degrees with the center of the circle.

Radii
Plural form of radius.

Radius
The segment whose endpoints are any point on a circle or sphere and its center; the length of that segment.

Secant to a circle
A line that intersects the circle in two points.

Sector
Part of a circle containing its center and an arc.

Semicircle
An arc whose central angle is a right angle.

Tangent to a circle
A line that intersects the circle in just one point, called point of tangency.

Interactive Geometric Applets: Relevant Theorems.

The angle formed by two secants intersecting outside the circle,

is determined by half the difference of the absolute value of the

two intersected arcs in between the secants.

Drag any of the movable points in this interactive

geometric applet to verify the relationship.

A secant and a tangent intersecting at an exterior point of the circle,

form an angle that is half the difference of the absolute value of the

intersected arcs, being these the in between the segments arcs. Drag any point

in the figure of the interactive applet to verify the relationship.

Be careful to keep the arcs at opposite sides of the circle.

Two tangents that intersect in the exterior of the circle, make an angle

that is half the difference of the absolute value of the two intercepted arcs in the circle.

Drag any point to verify this relationship in this interactive

geometric applet.

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