1. Circles and Chords
Exploring 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 the
are 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. The
are 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
decompressor
are 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are 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 are
the angles running
through the center
and two points on the QuickTimeª and a
decompressor
circle. They have the are needed to see this picture.
same measure as the
arcs 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between angles
are 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 tangency
Picture 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between angles
are needed to see this picture.
QuickTimeª and a
decompressor
are 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between angles
are needed to see this picture.
QuickTimeª and a
Relationships between chords and lines
decompressor
are needed to see this picture.
How to find the length of a chord
QuickTimeª and a
decompressor
are 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between angles
are needed to see this picture.
QuickTimeª and a
Relationships between chords and lines
decompressor
are needed to see this picture.
How to find the length of a chord
QuickTimeª and a
decompressor
are 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between chords and lines
are needed to see this picture.
QuickTimeª and a
How to find the length of a chord
decompressor
are needed to see this picture.
Relationships between angles
QuickTimeª and a
decompressor
are needed to see this picture.
How to find the measure of an arc
QuickTimeª and a
decompressor
are 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
decompressor
are needed to see this picture.
What is an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
Relationships between chords and lines
are needed to see this picture.
QuickTimeª and a
How to find the length of a chord
decompressor
are needed to see this picture.
Relationships between angles
QuickTimeª and a
decompressor
are needed to see this picture.
How to find the measure of an arc
QuickTimeª and a
decompressor
are needed to see this picture.
QuickTimeª and a
decompressor
How to find the length of an arc
are 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 enjoyed
Learning about circles
