Definitions and Circle Vocabulary As an introduction to the concepts in geometry first we’ll establish a vocabulary for the key aspects of circles.
Follow along with these definitions in geogebra using the link below. Visualizing the definitions

Radius is the measure from the center of a circle to the outside edge.

Diameter is the measure across a circle through its center. The diameter of a circle is twice its radius.

Pi or π is the measure of the circumference of a circle with a diameter of one.
It can also be understood as the circumference divided by the diameter of a circle.

Tangent is a line that intersects a circle at exactly one point.

Central Angle is an angle with endpoints located on a circle's circumference and vertex is located at the circle's center.

Inscribed Angle is an angle made from points sitting on the circle's circumference

Circumscribed Angle is an angle whose rays are tangent to the circle

From these simple definitions we can build a much more detailed understanding of the geometry of circles.
Together these definitions can help us understand some useful theorems related to circles.

Angle Inscribed in a Semicircle

Theorem 1: If an angle is inscribed in a semicircle, then the angle’s measure is one half the measure of the corresponding central angle

Angle at the Center Theorem: If an angles is inscribed in a circle, then the angle’s measure is one half the measure of the corresponding central angle.

TANGENT ANGLE THEOREM: The angle formed by a tangent line of a circle and the radius (or diameter) of the circle to the tangent point is always 90o. This means that if an inscribed or circumscribed triangle has the tangent line and radius or diameter as sides of the triangle, then Pythagorean rules and Pythagorean triples can be used to help solve problems of side length.

tangent.PNG

INTERSECTING CHORD THEOREM: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Looking at the example here, this means line segment a times line segment b is equal to line segment c times line segment d: ab=cd

chord intersection.PNG

INTERSECTING TANGENT LINES THEOREM: Two tangent lines to a circle that intersect at a single point outside the circle are equal in length.

tangent intersect.PNG

By extension, if a triangle is created by drawing a chord with endpoints at H and I, the resulting triangle is isosceles, and the angles at H and I are equal.

tangent intersect with chord.png

PRACTICE PROBLEMS FOR TANGENT, CHORD, AND SECANT __http://www.ixl.com/math/geometry/tangent-lines__ *simple practice questions using tangent rules (tangent to a circle, and two tangents intersecting at a point outside the circle)

__http://www.mathopenref.com/chordsintersecting.html__ *nice applet that lets you manipulate the length of the intersecting chords and shows that the product of the line segments produced by their intersection remains equal

Below are two manipulations that can be used to help visualize the intersecting chord theorem and the inscribed angle theorem, as well as other characteristics of circles. They were made so teachers could write-up their own lesson plan using the manipulations and tailor them to their students' needs.

## Resources for Circle Geometry Standards

Definitions and Circle VocabularyAs an introduction to the concepts in geometry first we’ll establish a vocabulary for the key aspects of circles.

Follow along with these definitions in geogebra using the link below.

Visualizing the definitions

Radiusis the measure from the center of a circle to the outside edge.Diameteris the measure across a circle through its center.The diameter of a circle is twice its radius.

Circumferenceis the distance around a circle.Unrolling a circle.

Circumference can be measured in many ways.

Piorπis the measure of the circumference of a circle with a diameter of one.It can also be understood as the circumference divided by the diameter of a circle.

More on what Pi is

Even more on Pi

Arcis a section of the circumference of a circle.Sectoris a slice of a circle.More on sectors

Chordis a line that links two points on a circleSecantis a line that intersects a circle twice.Tangentis a line that intersects a circle at exactly one point.Central Angleis an angle with endpoints located on a circle's circumference and vertex is located at the circle's center.Inscribed Angleis an angle made from points sitting on the circle's circumferenceCircumscribed Angleis an angle whose rays are tangent to the circleFrom these simple definitions we can build a much more detailed understanding of the geometry of circles.

Together these definitions can help us understand some useful theorems related to circles.

Angle Inscribed in a SemicircleTheorem 1: If an angle is inscribed in a semicircle, then the angle’s measure is one half the measure of the corresponding central angle

*If you need to edit proof:

Theorem 2: If an angle is inscribed in a semicircle, then the angle is a right angle.

*if need to edit proof:

Angle at the Center Theorem:If an angles is inscribed in a circle, then the angle’s measure is one half the measure of the corresponding central angle.

*If need to edit proof:

PRACTICE PROBLEMS FOR CENTRAL AND INSCRIBED ANGLESTANGENT ANGLE THEOREM:The angle formed by a tangent line of a circle and the radius (or diameter) of the circle to the tangent point is always 90o.This means that if an inscribed or circumscribed triangle has the tangent line and radius or diameter as sides of the triangle, then Pythagorean rules and Pythagorean triples can be used to help solve problems of side length.INTERSECTING CHORD THEOREM:If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.Looking at the example here, this means line segment

times line segmentais equal to line segmentbtimes line segmentc:dab=cdINTERSECTING TANGENT LINES THEOREM:Two tangent lines to a circle that intersect at a single point outside the circle are equal in length.By extension, if a triangle is created by drawing a chord with endpoints at H and I, the resulting triangle is isosceles, and the angles at H and I are equal.

PRACTICE PROBLEMS FOR TANGENT, CHORD, AND SECANT__http://www.ixl.com/math/geometry/tangent-lines__*simple practice questions using tangent rules (tangent to a circle, and two tangents intersecting at a point outside the circle)

__http://www.regentsprep.org/Regents/math/geometry/GP14/PracCircleTangents.htm__*more challenging practice questions using tangent rules

__http://regentsprep.org/Regents/math/geometry/GP14/PracCircleSegments.htm__*challenging problems involving chords, tangents, and secants

__http://www.mathopenref.com/chordsintersecting.html__*nice applet that lets you manipulate the length of the intersecting chords and shows that the product of the line segments produced by their intersection remains equal

## Below are two manipulations that can be used to help visualize the intersecting chord theorem and the inscribed angle theorem, as well as other characteristics of circles. They were made so teachers could write-up their own lesson plan using the manipulations and tailor them to their students' needs.

## Inscribed Angle Theorem Manipulation:

## http://tube.geogebra.org/student/m382547

## Intersecting Chord Theorem Manipulation:

## http://tube.geogebra.org/student/m382571

## Source and a very useful site for circles and Geometry in general:

## http://www.sparknotes.com/math/geometry2/theorems/section4.rhtml