Circles and Tangents in Geometry

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A series of free, online High School Geometry Lessons.
Examples, solutions, videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

  • about tangent lines to circles
  • about tangent segment to circles
  • inscribed angles
  • angles in semicircles and chords to tangents

Radii to Tangents

When a radius is drawn to a point of tangency, the angle formed is always a right (90 degree) angle. This fact is commonly applied in problems with two tangent segments drawn to a circle from a point. If two radii to tangents are drawn in, a kite with two right angles is formed and the missing angles or sides can be found. Related topics include central angles, tangent segments to a circle, and chords.
Basics of Tangent Lines to circles

This video provides example problems of determining unknown values using the properties of a tangent line to a circle.

Tangent Segments to a Circle

A tangent intersects a circle in exactly one point. When two segments are drawn tangent to a circle from the same point outside the circle, the segments are congruent. The extension problem of this topic is a belt and gear problem which asks for the length of belt required to fit around two gears. Topics related to circle radii include inscribed circles and radii to tangents.
How to find the length of tangent segments drawn to a circle from the same point?

This video shows how to solve for unknown values using the properties of tangent segments to a circle from a given point.

Inscribed Angles

Inscribed angles are angles whose vertices are on a circle and that intersect an arc on the circle. The measure of an inscribed angle is half of the measure of the intercepted arc and half the measure of the central angle intersecting the same arc. Inscribed angles that intercept the same arc are congruent.
How to calculate the measure of an inscribed angle.

Inscribed angles are 1/2 the measure of their intercepted arcs

Angles in Semicircles and Chords to Tangents

If an angle is inscribed in a semicircle, it will be half the measure of a semicircle (180 degrees), therefore measuring 90 degrees. Angles in semicircle is one way of finding missing missing angles and lengths. Pythagorean’s theorem can be used to find missing lengths (remember that the diameter is the hypotenuse). Also, the measure of an angle formed by a chord to a tangent is half the intercepted arc.
How to prove that an angle inscribed in a semicircle is a right angle; how to solve for arcs and angles formed by a chord drawn to a point of tangency.

A lesson on 1. the relationship of inscribed and tangent-chord angles with the same or congruent arcs
2. the measure of an inscribed angle whose intercepted arc is a semicircle
3. the relationship of tangent-tangent angles and their minor arcs

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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