Circles and Chords

Exploring the relationships between circumference,

diameter, chords, central angles,

and inscribed angles.

QuickTime™ and a decompressor

are needed to see this picture.

Photo from: http://www.shpefoundation.org/media/images/Escalante-photo.jpg

QuickTime™ and a decompressor

are needed to see this picture.

This is a diagram Showing the variousTypes of linesDrawn in relationTo a circle.We will use the Properties of theseLines to determine The measure and Length of arcs and Chords.

http://3.bp.blogspot.com/_ZMgCNR-NFuo/Swk7sD6DcEI/AAAAAAAAAJs/HiCFx4Q-xto/s320/CIRCLE_LINES.png

In this presentation you will learn:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

What is a chord

QuickTime™ and a decompressor

are needed to see this picture.

A line connecting two points on a circle is called a chord. The Chord AB connects the points A and B.

Photo from: http://www.graves.k12.ky.us/schools/gcms/academic_team/Academic%20Team%20Polygons%20Study%20Guide_files/image012.jpg

QuickTime™ and a decompressor

are needed to see this picture.

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

Check mark courtesy of: http://www.careersuccesstraining.com/images/CheckMark.jpg

What is an arc

An arc is the curve connecting two points on the circle.

The minor arc would be the red arc ABThe major arc would be the blue arc ACB

QuickTime™ and a decompressor

are needed to see this picture.

AB

C

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Relationships between angles

QuickTime™ and a decompressor

are needed to see this picture.

Picture from: http://jwilson.coe.uga.edu/EMAT6680/Huffman/InstructionalUnit/CentralAngle.jpg

Central angles are the angles running through the center and two points on the circle. They have the same measure as the arcs they intercept.

Intercepted Arc

Relationships between angles continued

Inscribed angles connect an arc to a point on the circle. Any inscribed angles intercepting the same arc have the same angle measure. Inscribed angles are half the measure of the central angle intercepting the same arc.

QuickTime™ and a decompressor

are needed to see this picture.

Picture from: http://upload.wikimedia.org/wikipedia/en/b/b5/Inscribed_angle_theorem.png

Angles Again…

QuickTime™ and a decompressor

are needed to see this picture.

http://www.algebra-answer.com/tutorials-2/greatest-common-factor/articles_imgs/7433/textbo18.jpg

Angle measures: a = 90˚ b = 90˚ c = 20˚ d = 200˚ e = 60˚ f = 120˚˚

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Relationships between chords and lines

A diameter can be drawn such that the diameter is a perpindicular bisector of the chord (It cuts the chord in half and the diameter and chord form right angles)

QuickTime™ and a decompressor

are needed to see this picture.

Picture from: http://www.exampaper.com.sg/blog/wp-content/uploads/2009/08/chord-perpendicular-bisector.gif

Relationships between chords and lines

A diameter can be drawn to form right angles to the tangent line at the point of tangency

QuickTime™ and a decompressor

are needed to see this picture.

Picture from: http://ocw.openhighschool.org/mod/resource/view.php?id=6522

QuickTime™ and a decompressor

are needed to see this picture.

Relationships between intersecting chords

Intersecting chords will create similar triangles

Picture from: http://www.winpossible.com/App_Themes/default/Images/CourseImages/Circle-Sector-Segments_Formed_by_Two_Intersecting_Chords.JPG

QuickTime™ and a decompressor

are needed to see this picture.

Angles AED and CEB are Vertical angles, and are Congruent. Angles DAB and BCD intersect the same arc And are therefore congruent.So by AA~, triangles DAB andBCD are congruent.

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

How to find the length of a chord:

First connect the points of the arc with radii

QuickTime™ and a decompressor

are needed to see this picture.

Picture 1 from: http://www.chiro.org/LINKS/GRAPHICS/IMAGE8.GIFPicture 2 from: http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/shirley1.1.gif

QuickTime™ and a decompressor

are needed to see this picture.Draw a perpindicular bisector,And now use a trig function suchAs sin(theta/2) to find one half ofThe chord’s length

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

So far we have learned:

What is a chordWhat is an arcRelationships between anglesRelationships between chords and

linesHow to find the length of a chordHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

How to find the measure of an arc

The measure of the arc is the same as the measure of the central angle or twice the measure of the inscribed angle.

QuickTime™ and a decompressor

are needed to see this picture.

So far we have learned:

What is a chordWhat is an arcRelationships between chords and

linesHow to find the length of a chordRelationships between anglesHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

How to find the length of an arc

The length of an arc is a part of the circumference of the circle.

Divide the central angle by 360˚ and times it by 2pi*r

QuickTime™ and a decompressor

are needed to see this picture.

So far we have learned:

What is a chordWhat is an arcRelationships between chords and

linesHow to find the length of a chordRelationships between anglesHow to find the measure of an arcHow to find the length of an arc

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Summary

A chord is a line connecting two points on a circle An arc is a curve connecting two points on a

circle A diameter can be drawn as a perpindicular

bisector of a chord An arc’s measure is the same as its intercepting

central angle, or twice its intercepting inscribed angle

Intersecting chords create similar triangles An arc’s length is found by forming triangles and

using trigonometry

Thank you for working through this tutorial.

I hope you have enjoyed Learning about circles

QuickTime™ and a decompressor

are needed to see this picture.