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.
|Slide Number 1||Hello everybody
Welcome to this tutorial on Theorems on Chords and Arcs in Geogebra
|Slide Number 2
|At the end of this tutorial,
you will be able to verify theorems on
|Slide Number 3
|We assume that you have the basic working knowledge of Geogebra.
For relevant tutorials, please visit our website
| Slide Number 4
|To record this tutorial I am using
Ubuntu Linux OS Version 11.10
Geogebra Version 18.104.22.168
|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|
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' >>un-check 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
|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
|Let's type value '3' for radius
|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 " circle-chord " in file name >> click on "Save"||Let us save this file now
Click on “File”>> "Save As"
I will type the file name as "circle-chord" click on “Save”
|Let us move on to the next 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|
|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
|In this tutorial, we have learnt to verify theorems on:
| 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
|Watch the video available at
/What is a Spoken Tutorial
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This is Madhuri Ganapathi from IIT Bombay signing off .
Thank you for joining