Cse684 circlesandchords

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A brief introduction to the relationships between arcs, chords, diameters, and central and inscribed angles.

A brief introduction to the relationships between arcs, chords, diameters, and central and inscribed angles.

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  • 1. Circles and ChordsExploring the relationships between circumference, diameter, chords, QuickTimeª and a decompressor are needed to see this picture. central angles, and inscribed angles. Photo from: http://www.shpefoundation.org/media/images/Escalante-photo.jpg
  • 2. This is a diagram Showing the various Types of lines Drawn in relation QuickTimeª and a To a circle. decompressor We will use theare needed to see this picture. Properties of these Lines 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
  • 3. In this presentation you will learn: What is a chord What is an arc Relationships between angles Relationships between chords and lines How to find the length of a chord How to find the measure of an arc How to find the length of an arc
  • 4. What is a chord QuickTimeª and a decompressor are needed to see this picture. A line connecting two points on QuickTimeª and a decompressor a circle is called a chord. Theare needed to see this picture. 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
  • 5. So far we have learned: QuickTimeª and a  What is a chord decompressorare needed to see this picture.  What is an arc  Relationships between angles  Relationships between chords and lines  How to find the length of a chord  How to find the measure of an arc  How to find the length of an arc Check mark courtesy of: http://www.careersuccesstraining.com/images/CheckMark.jpg
  • 6. What is an arc B A QuickTimeª and a decompressor are needed to see this picture. C An arc is the curve connecting two points on the circle. The minor arc would be the red arc AB The major arc would be the blue arc ACB
  • 7. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture.  Relationships between angles  Relationships between chords and lines  How to find the length of a chord  How to find the measure of an arc  How to find the length of an arc
  • 8. Relationships between anglesCentral angles arethe angles runningthrough the centerand two points on the QuickTimeª and a decompressorcircle. They have the are needed to see this picture.same measure as thearcs they intercept. Intercepted Arc Picture from: http://jwilson.coe.uga.edu/EMAT6680/Huffman/InstructionalUnit/CentralAngle.jpg
  • 9. Relationships between angles continued QuickTimeª and a decompressor are needed to see this picture.  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.Picture from: http://upload.wikimedia.org/wikipedia/en/b/b5/Inscribed_angle_theorem.png
  • 10. Angles Again…Angle measures: QuickTimeª and a 90˚ a decompressor = = 90˚ b c = 20˚ d = 200˚ are needed to see this 60˚ e = picture. f = 120˚˚ http://www.algebra-answer.com/tutorials-2/greatest-common-factor/articles_imgs/7433/textbo18.jpg
  • 11. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between anglesare needed to see this picture.  Relationships between chords and lines  How to find the length of a chord  How to find the measure of an arc  How to find the length of an arc
  • 12. Relationships between chords and lines A diameter can be drawn such that the diameter is a perpindicular QuickTimeª and a bisector of the decompressor chord (It cuts the are needed to see this picture. chord in half and the diameter and chord form right angles) Picture from: http://www.exampaper.com.sg/blog/wp-content/uploads/2009/08/chord-perpendicular-bisector.gif
  • 13. Relationships between chords and lines QuickTimeª and a decompressor are needed to see this picture. QuickTimeª and a decompressor are needed to see this picture.  A diameter can be drawn to form right angles to the tangent line at the point of tangencyPicture from: http://ocw.openhighschool.org/mod/resource/view.php?id=6522
  • 14. Relationships between intersecting chords  Intersecting chords will create similar triangles Angles AED and CEB are Vertical angles, and are QuickTimeª and a decompressor Congruent. Angles DAB and are needed to see this picture. BCD intersect the same arc And are therefore congruent. So by AA~, triangles DAB and BCD are congruent.Picture from: http://www.winpossible.com/App_Themes/default/Images/CourseImages/Circle-Sector-Segments_Formed_by_Two_Intersecting_Chords.JPG
  • 15. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between anglesare needed to see this picture. QuickTimeª and a decompressorare needed to see this picture.  Relationships between chords and lines  How to find the length of a chord  How to find the measure of an arc  How to find the length of an arc
  • 16. How to find the length of a rc with radii chord: s of the a the point con nect First QuickTimeª and a decompressor are needed to see this picture. QuickTimeª and a decompressor are needed to see this picture. Draw a perpindicular bisector, And now use a trig function such As sin(theta/2) to find one half of The chord’s length Picture 1 from: http://www.chiro.org/LINKS/GRAPHICS/IMAGE8.GIF Picture 2 from: http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/shirley1.1.gif
  • 17. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between anglesare needed to see this picture. QuickTimeª and a  Relationships between chords and lines decompressorare needed to see this picture.  How to find the length of a chord QuickTimeª and a decompressorare needed to see this picture.  How to find the measure of an arc  How to find the length of an arc
  • 18. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between anglesare needed to see this picture. QuickTimeª and a  Relationships between chords and lines decompressorare needed to see this picture.  How to find the length of a chord QuickTimeª and a decompressorare needed to see this picture.  How to find the measure of an arc  How to find the length of an arc
  • 19. How to find the measure of an arc The measure of the arc is the same as the measure of the central angle or QuickTimeª and a decompressor are needed to see this picture. twice the measure of the inscribed angle.
  • 20. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between chords and linesare needed to see this picture. QuickTimeª and a  How to find the length of a chord decompressorare needed to see this picture.  Relationships between angles QuickTimeª and a decompressorare needed to see this picture.  How to find the measure of an arc QuickTimeª and a decompressorare needed to see this picture.  How to find the length of an arc
  • 21. How to find the length of an arc QuickTimeª and a decompressor are needed to see this picture. 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
  • 22. So far we have learned:  What is a chord QuickTimeª and a decompressorare needed to see this picture.  What is an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  Relationships between chords and linesare needed to see this picture. QuickTimeª and a  How to find the length of a chord decompressorare needed to see this picture.  Relationships between angles QuickTimeª and a decompressorare needed to see this picture.  How to find the measure of an arc QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor  How to find the length of an arcare needed to see this picture.
  • 23. 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
  • 24. Thank you for working through this tutorial. QuickTimeª and a decompressor are needed to see this picture.I hope you have enjoyedLearning about circles