Geogebra/C3/TheoremsonChordsandArcs/English
Title of script: Theorems on Chords and Arcs
Author: Madhuri Ganapathi
Keywords: Geogebra tools used Circle with Center and Radius, Circular Sector with Center between Two Points, Circular Arc with Center between Two points, Midpoint, Angle, Perpendicular line, Polygon, Intersect Two objects and Distance or Length
Object Properties, Chord, Arcs, Graphic view, Axes, Grid Spoken tutorial, video tutorial
Note to Translators  Translators do no translate theorems , please refer to standard mathematics text books of classes IX and X of your language for the original version of the theorems.
Visual Cue  Narration 

Slide Number 1  Hello everybody
Welcome to this tutorial on Theorems on Chords and Arcs in Geogebra 
Slide Number 2
Learning Objectives 
At the end of this tutorial,
you will be able to verify theorems on

Slide Number 3
Prerequisites 
We assume that you have the basic working knowledge of Geogebra.
If not, For relevant tutorials, please visit our website 
Slide Number 4
System Requirement 
To record this tutorial I am using
Ubuntu Linux OS Version 11.10 Geogebra Version 3.2.47.0 
Slide Number 5
GeoGebra Tools used in this tutorial 
We will use the following Geogebra tools

Switch to GeoGebra window
Dash home >>Media Apps>>Under Type choose>>Education>>Geogebra 
Let's open a new GeoGebra window.
Click on Dash home Media Apps. Under Type Choose Education and GeoGebra. 
Let's state a theorem  
Slide 6
Theorem Show a complete glimpse of the end product. 
Perpendicular from center of circle to a chord bisects the chord
Perpendicular from center A of a circle to chord BC bisects 
Let's now verify the theorem.  
Right Click on the drawing pad >>
'Graphic view' >>uncheck on 'Axes'>> Select Grid 
For this tutorial I will use 'Grid layout' instead of Axes
Right Click on the drawing pad In the 'Graphic view' uncheck the 'Axes' and Select 'Grid' 
Click on “Circle with center and Radius”>>click on point A  Let's draw a circle.
Select the "Circle with Center and Radius" tool from the tool bar. Mark a point 'A' on the drawing pad. 
A dialogue box open  A dialogue box open 
Type value '3' for radius
Click OK 
Let's type value '3' for radius
Click OK. 
Point to the circle
Move point 'A' to show the motion of the circle. 
A Circle with center 'A' and radius '3cm' is drawn
Let's Move point 'A' and see the movement of the circle. 
Click on “Segment between two points” tool>>mark points 'B' and 'C'  Select “Segment between two points” tool.
Mark points 'B' and 'C' on the circumference of the circle A chord 'BC' is drawn. 
Click on "Perpendicular line" tool
>>click on Segment 'BC'>>Click on 'A' 
Let's draw a perpendicular line to Chord 'BC' which passes through 'A'.
Click on "Perpendicular line" tool from tool bar Click on chord 'BC', and point 'A'. 
Move point 'B'>> perpendicular line moves along with point 'B'  Let's Move point 'B', and see how the perpendicular line moves along with point 'B'. 
Point to the intersection point  Perpendicular line and Chord 'BC' intersect at a point 
Select “Intersect Two objects” tool>>mark point of intersection as 'D'  Click on “Intersect Two objects” tool,
Mark the point of intersection as 'D'. 
Let's verify whether D is the mid point of chord BC  
Click on "Distance or Length" tool>>measure BD and CD
Cursor on the distance measure 
Click on "Distance or Length" tool... Click on the points , 'B' and 'D' ... 'D' and 'C' ... Notice that distances 'BD' and 'DC' are equal. It implies 'D' is midpoint of chord 'BC' 
Click on angle tool >>Measure angle 'ADC'
Use the dynamic nature of the pad. 
Let's measure angle 'CDA'
Click on Angle tool ... Click on the points 'C','D', 'A' Notice that angle 'CDA' is '90^0'. 
Move point 'C'  Let's Move the point 'C'
and see how the distances move along with point 'C' 
Theorem is thus verified.  
Click on "Save As" >> type " circlechord " in file name >> click on "Save"  Let us save this file now
Click on “File”>> "Save As" I will type the file name as "circlechord" click on “Save” 
Let us move on to the next theorem.  
Slide 8
Theorem Visual teaser of the theorem needed 
Inscribed angles subtended by the same arc are equal.
Inscribed angles BDC and BEC subtended by the same arc BC are equal 
Let's verify the theorem.  
Click on “File” >> "New"  Open a new Geogebra window,
Click on “File” >> "New" 
Click on "Circular Arc with Center between Two points" tool>>
Click on point A'>>mark points 'B' and 'C' on the circumference 
Let's draw an arc 'BC'
Click on "Circular Arc with Center between Two points" Click on point 'A' then click on points 'B' and 'C' on the circumference An Arc 'BC' is drawn 
Point to "Algebra view"
Right click on object 'd'>> Select "Object Properties" >> Select color>> 'Green' Click on "Close" 
Let's change properties of arc 'BC'
In "Algebra View" Right click on object 'd' Select "Object Properties" Select Color as 'Green' Click on "Close" 
Click on "New point" tool >>Mark points 'D' and 'E' on circumference  Click on "New point" tool.
Mark points 'D' and 'E' on circumference of circle. 
Click on "Polygon" tool>>click on points 'C', 'B', 'D', 'E','C'  Let's subtend two angles from the arc 'BC' at points 'D' and 'E'.
Click on "Polygon" tool, then click on points 'E', 'B', 'D', 'C' and 'E' to complete the figure. 
Click on "Angle" tool>> click on points 'C', 'D', 'B'  Let's measure the angles 'BDC' and 'BEC'
Click on "Angle" tool, Click on points 'B', 'D', 'C' and 'B', 'E', 'C' 
Point to the angles  We can see that the angles 'BDC' and 'BEC' are equal. 
Let's state a next theorem  
Slide 9
Theorem 
Angle subtended by an arc at the center, is twice the inscribed angles subtended by the same arc
Angle BAC subtended by arc BC at A is twice the inscribed angles BEC and BDC subtended by the same arc 
Let's verify the theorem  
Click on "Circular Sector with Center between Two Points" tool>>draw sector  Let's draw a sector 'ABC'
Click on "Circular Sector with Center between Two Points" tool. click on points 'A', 'B', 'C' 
Select sector 'ABC'>>Right click>>select"Object Properties">>
Object properties window opens>>select color “Green” click on "Close" 
Let's change the color of the sector 'ABC'.
Right click on sector 'ABC' Select "Object Properties". Select Color as “Green”. Click on "Close". 
Click on "Angle" tool>> measure angle 'BAC'  Let's the measure angle 'BAC'
Click on "Angle" tool Click on the points 'B', 'A', 'C' Angle 'BAC' is twice the angles 'BEC' and 'BDC' 
Point to the angles
Dynamic visuals needed. 
Let's move point 'C'
Notice angle 'BAC' is always twice the angles 'BEC' and 'BDC' hence theorems are verified 
With this we come to the end of the tutorial
let's summarise  
Slide 9
Summary 
In this tutorial, we have learnt to verify theorems on:

Slide 10
Assignment 
As an assignment I would like you to verify
Equal chords of a circle are equidistant from center. Draw a circle. Select Segment with Given length from point tool Use it to draw two chords of equal size. Draw perpendicular lines from center to chords. Mark points of intersection. Measure perpendicular distances. 
Show the output of the Assignment  The out put should look like this 
Slide number 11
Acknowledgement 
Watch the video available at
/What is a Spoken Tutorial It summarises the Spoken Tutorial project If you do not have good bandwidth, you can download and watch it 
The Spoken Tutorial Project Team :
Conducts workshops using spoken tutorials Gives certificates to those who pass an online test For more details, please write to contact@spokentutorial.org  
Spoken Tutorial Project is a part of the Talk to a Teacher project
It is supported by the National Mission on Education through ICT, MHRD, Government of India More information on this Mission is available at http://spokentutorial.org/NMEICTIntro ] This is Madhuri Ganapathi from IIT Bombay signing off . Thank you for joining 