Geogebra/C3/Theorems-on-Chords-and-Arcs/English

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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

Click here for Slides

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

  • Chords of circle.
  • Arcs of circle.
Slide Number 3

Pre-requisites

We assume that you have the basic working knowledge of Geogebra.

If not,

For relevant tutorials, please visit our website

http://spoken-tutorial.org

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
  • Circle with Center and Radius
  • Circular Sector with Center between Two Points
  • Circular Arc with Center between Two points
  • Midpoint and
  • Perpendicular line
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' >>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

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 " 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.
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:
  • Perpendicular line from center to chord bisects it
  • Inscribed angles subtended by the same arc are equal
  • the Central angle of a circle is twice any inscribed angle subtended by the same arc
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

http://spoken-tutorial.org

/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@spoken-tutorial.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://spoken-tutorial.org/NMEICT-Intro ]

This is Madhuri Ganapathi from IIT Bombay signing off .

Thank you for joining

Contributors and Content Editors

Chandrika, Madhurig