Madhurig at 11:53, 7 December 2021

2021-12-07T11:53:21Z

Madhurig: Created page with "{| border=1 || '''Visual Cue''' || '''Narration''' |- || '''Slide Number 1''' '''Title Slide''' || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGeb..."

2021-11-12T07:32:50Z

Created page with "{| border=1 || '''Visual Cue''' || '''Narration''' |- || '''Slide Number 1''' '''Title Slide''' || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGeb..."

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{| border=1
|| '''Visual Cue'''
|| '''Narration'''
|-
|| '''Slide Number 1'''

'''Title Slide'''
|| Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGebra'''.
|-
|| '''Slide Number 2'''

'''Learning Objectives'''

|| In this tutorial, we will learn about the properties of,
* Chords
* Arcs and sectors and
* Tangents

|-
||'''Slide Number 3'''

'''System Requirement'''
|| To record this tutorial, I am using;

'''Ubuntu Linux''' OS version 18.04

'''GeoGebra''' version 5.0.660.0-d
|-
||'''Slide Number 4'''

'''Pre-requisites'''

'''https://spoken-tutorial.org'''
||To follow this tutorial, learner should be familiar with '''GeoGebra''' interface.

For the prerequisite '''GeoGebra''' tutorials please visit this website.
|-
||Cursor on the '''GeoGebra''' window.
|| I have opened a new '''GeoGebra '''window.
|-
|| Right-click on the Graphics view.

'''Graphic view''' window opens >> un-check the '''Axes''' check box.
||Let us uncheck the '''Axes'''.

Right-click in the '''Graphics view'''.

In the '''Graphics''' menu, uncheck the '''Axes''' check box.
|-
||Point to the '''Algebra View'''.

Click on the '''Toggle Style Bar''' arrow.

In menu click on '''Sort by''' drop-down select '''Object Type''' check box.
||In the'''Algebra view''' click on the '''Toggle Style Bar ''' arrow.

In the '''Sort by''' drop-down, select '''Object Type''' check box, if not already selected.
|-
|| Cursor on the interface.
|| Let us now learn about the property of a chord.
|-
|| '''Slide Number 5'''

'''Properties of Chords'''

Show the glimpse of the completed figure.
|| It states that -

Perpendicular from the centre of a circle to a chord bisects the chord.

|-
||Select the '''Circle: Center & Radius''' tool from the tool bar.

Click in the '''Graphics view''' to mark a point '''A''' in the '''Graphics view'''.
|| Let us draw a circle.

Select the '''Circle: Center & Radius''' tool from the tool bar.

Click in the '''Graphics view''' to mark a point '''A'''.
|-
||Point to the text box '''Circle: Center & Radius'''.
||'''Circle: Center & Radius''' text box opens.
|-
|| Type value 3 for radius

Click '''OK''' button.
||In the '''Radius''' field let us type 3 and click the '''OK '''button.

|-
||Point to the circle '''c'''.

Point to point '''A'''.

||A circle '''c''' with centre '''A''' and radius 3 centimetres is drawn in the '''Graphics view'''.

|-
||Click on the '''Segment ''' tool >> mark points '''B''' and '''C''' on the circumference.

Point to the chord '''BC'''.
||Select the '''Segment''' tool.

Click to mark two points '''B''' and '''C ''' on the circumference as shown.

Chord '''BC''', named as '''f''' is drawn on the circle '''c'''.
|-
||Click on '''Perpendicular Line''' tool.

Click on Segment '''BC''' >> Click on '''A'''.
||Let’s drop a perpendicular line to chord '''BC''' passing through '''A'''.

Click on the '''Perpendicular Line''' tool.

Click on chord '''BC''', and then on point '''A'''.

|-
||Click on the '''Move''' tool.

Move point '''B''' >> perpendicular line moves along with point '''B'''.
||Let us move point '''B'''.

Observe that the perpendicular line moves along with point '''B'''.
|-
|| Point to the intersection point.

Select '''Intersect''' tool >> mark point of intersection as '''D'''.
|| The perpendicular line and chord '''BC''' intersect at a point.

Using the '''Intersect''' tool let’s mark the intersection point as '''D'''.
|-
|| Cursor on '''D'''.

Point to the values in the '''Algebra View'''.
||Let’s measure the lengths '''BD''' and '''DC'''.
|-
||Click on '''Distance or Length''' tool >> measure '''BD''' and '''DC'''.

||Click on the '''Distance or Length ''' tool.

Click on the points, '''B''' and '''D''' and then '''D''' and '''C'''.
|-
|| Point to the values in the '''Algebra View'''.

Cursor on the distance measure.

Highlight the values in the '''Algebra View'''.
||Notice that distances '''BD''' and '''DC''' are equal.

It implies that '''D''' is midpoint of chord '''BC'''.

Note that the perpendicular from the centre '''A''' to chord '''BC''' bisects it.

|-
|| Click on the '''Move''' tool >> Drag the labels.
||Let us move all the labels using the '''Move''' tool to see them clearly.
|-
||Click on '''Angle''' tool >> Click on the points '''C''', '''D''', '''A''' in the anticlockwise.

Point to the angle.

|| Now let’s measure the angle '''CDA'''.

Click on '''Angle''' tool and click the points '''C''', '''D''' and '''A'''.

Angle '''CDA''' is 90<sup>0</sup> degrees.

A line drawn from the centre to the midpoint of the chord is perpendicular to it.

|-
|| Click on the '''Move''' tool >> Drag point '''C'''.
|| Let us move point '''C''' and see how the distances change accordingly.
|-
||'''Slide Number 5 + 6'''

'''Assignment'''

|| Pause the tutorial and do this assignment.

Open a new '''GeoGebra''' window.

Draw a circle.

Draw two chords of equal size to the circle.

Draw perpendicular lines from the centre to the chords.

Mark points of intersection.

Measure the perpendicular distances.

What do you observe?

|-
|| Show the glimpse of the assignment.
|| The completed assignment should look like this.

Observe that, equal chords of a circle are equidistant from centre.

|-
|| Cursor on the '''GeoGebra''' window.

Point to '''c'''.

Point to '''A''', '''B''', and '''C'''.

||Now let us go back to the circle.

Let us retain circle '''c''' and points '''A''', '''B''' and '''C'''.

Delete the rest of the objects.
|-
||Go to '''Algebra View''' >> Press '''Ctrl''' key >> click to select the objects.

Press '''Delete '''key on the Keyboard.

||Go to the '''Algebra view'''.

Press the '''Ctrl''' key and select the objects for deletion.

Then press '''Delete''' key on the keyboard.
|-
|| Show the glimpse of the completed figure at the time of recording.
||Next let us prove a property with respect to an '''arc'''.

Inscribed angles '''BDC''' and '''BEC''' subtended by the same '''arc BC''' are equal.

|-
||Click on '''Circular Arc''' tool.

Click on point '''A''' >> Click on points '''B''' and '''C''' on the circumference.

Point to '''arc d'''.
||Let us next draw an '''arc'''.

Click on the '''Circular Arc''' tool.

Click on point '''A'''.

Then click on points '''B''' and '''C''' on the circumference.

An '''arc d''' is drawn.
|-
||Point to '''Algebra view'''.

Right click on object '''d'''.

Select '''Object Properties'''.

||Let us change properties of '''arc d'''.

In the '''Algebra View''', right-click on object '''d'''.

Select '''Object Properties''' from the '''context menu'''.
|-
||Point to the '''Properties''' window.

Click on the '''Color''' tab and select the colour as green.

||'''Properties''' window opens next to '''Graphics view'''.

Click on the '''Color''' tab and select green colour.
|-
|| Click on the '''Style''' tab.

In the '''Filling''' drop-down, select ''' Hatching'''.
|| Let us change the style of filling of the '''arc d'''.

Select the '''Style''' tab and change the '''Filling''' to '''Hatching'''.
|-
|| Click on the '''X''' icon.
|| Close the '''Properties''' window.

|-
||Click on '''Point''' tool >> Mark points '''D''' and '''E''' on circumference.

Mark the point '''D''' above point '''B''' and point '''E''' above point '''C'''.
|| Let us mark two points on the circumference of the circle.

Click on '''Point''' tool.

Mark point '''D''' above point '''B''' and point '''E''' above point '''C'''.

|-
|| Cursor on the points.
||Let us subtend two angles from '''arc BC''' to points '''D''' and '''E'''.

|-
||Select the '''Segment''' tool and Join the points

'''B,E''' '''E,C''' '''B,D''' '''D,C'''.
||Select the '''Segment''' tool and join the following points.

'''B,E''' '''E,C''' '''B,D''' and '''D,C'''.
|-
||Click the '''Angle''' tool.

Click the segments '''BD''', '''DC'''.

Click the Segments '''BE''', '''EC'''.
|| Let’s measure the angles '''BDC''' and '''BEC'''.

Click on the '''Angle''' tool,

Click the segments that form the angle.

'''BD''' and '''DC''' and then click '''BE''' and '''EC'''.
|-
|| Point to the angles

Highlight the measure of the angles in the '''Algebra view'''.
||Observe that the angles '''BDC''' and '''BEC''' are equal.

This proves the property that angles formed using the same '''arc''' are equal.
|-
||Click on '''Circular Sector''' >> draw a '''sector'''.

Click the points '''A''', '''B''', '''C'''.
||Let’s draw a sector '''ABC'''.

Click on '''Circular Sector'''tool.

Now click the points '''A''', '''B''', and '''C'''.

Sector '''ABC''' is drawn.
|-
||Click on '''Angle''' tool.

Click on the points '''B''', '''A''', '''C'''.

Point to the angles '''BDC''' and '''BEC'''.

||Let’s measure the angle '''BAC''' using the '''Angle''' tool.

Observe that angle '''BAC''' is twice the angles '''BDC''' and '''BEC'''.
|-
||Click on '''Move''' tool >> drag point '''C'''.

Point to the angles in the '''Algebra view'''.

Point to angles '''BEC''' and '''BDC'''.

Point to the angle '''BAC'''.
|| Using the '''Move''' tool let’s move point '''C''' to change the angles.

Notice the angles '''BEC''' and ''' BDC''' subtended by the '''arc d'''.

Angle '''BAC''' is always twice the angles subtended by the '''arc d'''.

Here angle at the centre is twice any inscribed angle subtended by the same '''arc'''.

|-
|| Cursor on the interface.
||Next let us construct a pair of tangents to a circle.
|-
|| Click on '''File''' >> Select '''New Window'''.
|| Let us open a new '''GeoGebra''' window.
|-
|| Right-click on the '''Graphics view'''.

'''Graphic view''' window opens

>> un-check the '''Axes''' check box.
||Let us uncheck the '''Axes'''.
|-
||Select '''Circle: Center & Radius''' tool.

Click in the '''Graphics view''' to mark point '''A'''.
||Let's draw a circle using '''Circle: Center & Radius''' tool.

Click in the '''Graphics view''' to mark point '''A'''.
|-
||'''Circle: Center & Radius''' text box opens.

Type 3 for radius in the '''Radius''' field.

Click '''OK''' button.

Click on '''Move''' tool and move point '''A'''.
|| Type 3 for radius in the text box that opens.

Then click '''OK''' button.

A circle '''c''' with centre '''A''' and radius 3 centimetres is drawn.
|-
||Click on '''Point''' tool >> Mark point '''B'''.
||Now click on the '''Point''' tool.

Click to mark a point '''B''' outside the circle.
|-
||Select ''' Segment ''' tool >> join points '''A''' and '''B'''.
|| Using the '''Segment''' tool join points '''A''' and '''B''' to draw segment '''f'''.

|-
||Select '''Perpendicular Bisector''' tool >> click point '''A''' and point '''B'''.

||Let us draw a perpendicular bisector to segment '''f'''.

Select the '''Perpendicular Bisector''' tool, click on points '''A''' and '''B'''.

|-
||Click on '''Intersect''' tool >> Point '''C'''.
||Segment '''f''' and perpendicular bisector intersect at a point.

Click on '''Intersect''' tool to mark the point of intersection as '''C'''.

|-
||Click on '''Move''' tool and move point ''' B'''.
|| Let's move point B.

Observe that perpendicular bisector and point '''C''' move along with point '''B'''.

This is because these objects are dependent on point ''' B'''.
|-
||'''Slide Number 7'''

'''Assignment'''
|| Pause the tutorial and do this assignment.

Verify if point '''C''' is the midpoint of segment '''f'''.
|-
|| Cursor on the interface.
|| Now let us draw another circle.
|-
||Select ''' Compass''' tool from tool bar

>> Click Point '''C'''

>> Point '''B''' >> Point '''C'''.
||Select the '''Compass''' tool.

Click on the points '''C''', '''B''' and '''C''' again

to complete the figure.
|-
|| Point to the two points of intersection.

Click on '''Intersect''' tool >> mark point '''D'''

>> mark point '''E'''.
|| Two circles intersect at two points.

Using the ''' Intersect''' tool, mark the points of intersection as '''D ''' and '''E'''.
|-
|| Select '''Segment''' tool

>> Join '''B''' and '''D '''

>> Join '''B''' and ''' E'''.
|| Select the '''Segment''' tool.

Join the points '''B''', '''D''' and '''B''', '''E'''.
|-
|| Point to segments '''h''' and '''i'''.

Point to the circle '''c'''.
|| Segments '''h''' and ''' i''' are the tangents to circle '''c'''.
|-
||
|| Let's explore some more properties of the tangents to the circle.
|-
||Click '''Segment''' tool

>> join '''A''', '''D'''

>> join '''A''', '''E'''.
||Using the '''Segment''' tool and join the points '''A''', '''D''' and '''A''', '''E'''.
|-
||Outline the triangles '''ABD''' and '''ABE'''.

Point to Segments '''AD = j ''' and '''AE = k'''.

Point to the values in ''' Algebra view.'''

|| Let us show that triangles '''ABD''' and '''ABE''' are congruent.

Segment '''j''' is equal to segment '''k''', as they are radii of circle '''c'''.

In the '''Algebra view''' observe that segment '''j''' is equal to segment '''k'''.
|-
|| Point to angles '''ADB''' and '''BEA'''

Outline the semicircle '''d'''.
|| Angle '''ABD''' is equal to angle ''' BEA''' ('''∠ADB''' = '''∠BEA''').

As they are angles on the semicircles of the circle '''d'''.

Let’s measure the angles.
|-
|| Click '''Angle''' tool.

Click the segments that make the angles.

Click the segments '''j''' and '''h''' to measure the angle.

Click the segments '''i''' and '''k''' to measure the angle.

Point to the angles.
|| Select the '''Angle''' tool.

Click the segments '''j''', '''h''' and '''i''', '''k''' to measure the angles.

Notice that are equal and 90<sup>0 </sup> degrees.
|-
|| Point to segment '''f'''.

Outline the two triangles.

'''△ABD''' is congruent to(≅) '''△ABE''' by '''SAS''' rule of Congruence.
||Segment '''f''' is the common side for both the triangles.

Therefore triangle '''ABD''' is congruent to triangle '''ABE''' by '''SAS''' rule of '''Congruence'''.
|-
|| Point to tangents '''BD''' and '''BE.'''
|| It implies that tangents '''BD''' and '''BE''' are equal.
|-
|| Point to the Algebra view.

'''BD =h ''' and '''BE= i'''.
|| From the '''Algebra view''',

observe that segments '''h''' and '''i''' are equal.
|-
|| Point to the angle, radius, tangent.
|| Tangents are perpendicular to the radius of the circle at the point of contact.
|-
|| Click on '''Move''' tool >> move point '''B'''.
||Let's move point '''B''' and see how the tangents move along with point '''B'''.

Tangents are drawn from point ''B''', so they are dependent on it.
|-
|| Right-click on point ''' B'''.

From the context menu select '''Delete'''.

Point the place.
|| Let’s now delete point '''B'''.

Right-click on point '''B''', from the context menu select '''Delete'''.

All the objects dependent on point '''B''' are deleted along with it.
|-
|| Point to the circle.
|| We now have a circle '''c''' with centre '''A''' on the '''Graphics view'''.
|-
||Click on '''Point''' tool

>> click point '''B'''

>> click point '''C''' on the circumference.

Mark point '''D''' outside the circle.
||Select the ''' Point''' tool.

Mark points '''B''' and '''C''' on the circumference and '''D''' outside the circle.
|-
||Click '''Tangents''' tool

>> Click on point '''D '''

>> Click on the circumference.

||Select the '''Tangents''' tool.

Click on point '''D''' and then on the circumference.

Two Tangents are drawn to the circle '''c'''.
|-
||Point to the points of contact.

Click on '''Intersect''' tool

>> click on the intersection points.
|| Tangents meet at two points on the circle.

Click on the '''Intersect''' tool and mark points of contact as '''E''' and '''F'''.
|-
|| Click on '''Polygon''' tool.

Click on point '''B'''

>> point '''C''' >> point '''F''' >> point '''B''' again to complete the figure.
|| Let us draw a triangle.

Click on the '''Polygon''' tool.

Click on the points '''B''', '''C''', '''F''' and '''B''' again to complete the figure.
|-
||Point to the segment '''CF'''.
||In the figure segment '''b''' is the chord to the circle '''c'''.
|-
||Point to '''∠FCB''' and the chord.
||Angle '''FBC (∠FBC)''' is the inscribed angle by the chord '''CF''' to the circle '''c'''.
|-
|| Point '''∠DFB''' and tangent.
||Angle '''DFC(∠DFC)''' is the angle between tangent and chord to circle '''c'''.
|-
||Click on '''Angle''' tool.

Click on the points '''F''', '''B''', '''C'''

>> click on the points '''D''', '''F''', '''C'''.
|| Let’s measure the angles.

Click on the '''Angle''' tool.

Click on the points '''F''', '''B''', '''C''' and '''D''', '''F''', '''C'''.
|-
|| Point the angles.

Point to the angle in the '''Algebra View'''.

Point to the angle '''DFC'''.

Point to the angle '''FBC''' and '''chord CF'''.
||Notice that angle '''DFC''' is equal to angle '''FBC '''('''∠DFC =∠FBC''').

Angle '''DFC''' is the angle between tangent and chord '''CF'''.

This angle is equal to inscribed angle '''FBC''' of the chord '''CF'''.

|-
||Click on '''Move''' tool and move point '''D'''

Point to point '''D'''.
||Let's move point '''D'''.

'''Observe''' that tangents and chord '''CF''' move along with point '''D'''.

Here all the objects are dependent on point '''D''' as the tangents are drawn from it.
|-
||Click on '''File ''' >> '''Save As'''.

Point to the '''Save''' dialog box.

Point the '''Desktop ''' folder.

Type the file name as '''Tangents'''

Click on '''Save ''' button.
|| Let us save this file now

Click on '''File ''' then ''' Save'''.

I will save the file on the ''' Desktop'''.

In the '''Save''' dialog box type the file name as '''Tangents'''.

Click on '''Save ''' button.

|-
||
|| With this, we come to the end of the tutorial.

Let us summarise.
|-
||'''Slide Number 7'''

'''Summary '''
|| In this tutorial, we have learnt about the properties of,
* Chords
* Arcs and sectors and
* Tangents

|-
||'''Slide Number 8'''

'''Assignment'''
|| As an assignment.

Open a new '''GeoGebra ''' window.

Draw a circle.

Draw tangents from an external point.

Mark points of intersection of the tangents.

Join the centre of the circle to intersection points

Measure angle at the centre and measure angle between the tangents.

What is the sum of the two angles?

Join the centre and the external point.

Does the line segment bisect the angle at the centre?

|-
|| Show the output of the Assignment
|| The output of the assignment should look like this.

|-
|| '''Slide Number 9'''

'''About Spoken Tutorial Project'''
||
* The video at the following link summarises the Spoken Tutorial project.
* Please download and watch it

|-
|| '''Slide Number 10'''

'''Spoken tutorial workshops'''
||
* We conduct workshops using Spoken Tutorials and give certificates.
* For more details, please contact us.

|-
|| '''Slide''' '''Number 11'''

'''Forums'''
|| Please post your timed queries in this forum.
|-
|| '''Slide Number 12'''

'''Acknowledgement'''

|| The '''Spoken Tutorial''' project is funded by the '''Ministry of Education '''Govt. of India.
|-
||
|| This is Madhuri Ganapathi from, IIT Bombay signing off.

Thank you for watching.
|-
|}

@@ Line 26: / Line 26: @@
 '''GeoGebra''' version 5.0.660.0-d
 |-
 ||'''Slide Number 4'''

GeoGebra-5.04/C3/Properties-of-Circles/English - Revision history

Madhurig at 11:53, 7 December 2021

Madhurig: Created page with "{| border=1 || '''Visual Cue''' || '''Narration''' |- || '''Slide Number 1''' '''Title Slide''' || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGeb..."