PoojaMoolya: Created page with "{| border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGebra'''. |- ||00:07 || In this tutorial,..."

2022-03-29T09:51:14Z

Created page with "{| border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGebra'''. |- ||00:07 || In this tutorial,..."

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{| border=1
|| '''Time'''
|| '''Narration'''
|-
|| 00:01
|| Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGebra'''.

|-
||00:07
|| In this tutorial, we will learn about the properties of,

Chords

|-
||00:12
|| Arcs and sectors and

Tangents

|-
||00:16
|| To record this tutorial, I am using;

|-
||00:19
||'''Ubuntu Linux''' OS version 18.04

|-
||00:24
||'''GeoGebra''' version 5.0.660.0-d

|-
||00:31
||The steps demonstrated in this tutorial will work exactly the same in lower versions of '''GeoGebra'''.

|-
||00:39
||To follow this tutorial, learner should be familiar with '''GeoGebra''' interface.

|-
||00:45
||For the prerequisite '''GeoGebra''' tutorials please visit this website.
|-
||00:50
|| I have opened a new '''GeoGebra '''window.

|-
|| 00:54
||Let us uncheck the '''Axes'''.

|-
||00:57
||Right-click in the '''Graphics view'''.

|-
||01:00
||In the '''Graphics''' menu, uncheck the '''Axes''' check box.

|-
||01:05
||In the'''Algebra view''' click on the '''Toggle Style Bar ''' arrow.

|-
||01:10
||In the '''Sort by''' drop-down, select '''Object Type''' check box, if not already selected.
|-
|| 01:17
|| Let us now learn about the property of a chord.
|-
|| 01:21
|| It states that - Perpendicular from the centre of a circle to a chord bisects the chord.

|-
||01:28
|| Let us draw a circle.

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

|-
||01:36
||Click in the '''Graphics view''' to mark a point '''A'''.
|-
||01:40
||'''Circle: Center & Radius''' text box opens.
|-
|| 01:45
||In the '''Radius''' field let us type 3 and click the '''OK '''button.

|-
||01:50
||A circle '''c''' with centre '''A''' and radius 3 centimetres is drawn in the '''Graphics view'''.

|-
||01:57
||Select the '''Segment''' tool.

|-
||02:00
||Click to mark two points '''B''' and '''C ''' on the circumference as shown.

|-
||02:06
||Chord '''BC''', named as '''f''' is drawn on the circle '''c'''.

|-
||02:11
||Let’s drop a perpendicular line to chord '''BC''' passing through '''A'''.

|-
||02:16
||Click on the '''Perpendicular Line''' tool.

|-
||02:20
||Click on chord '''BC''', and then on point '''A'''.

|-
||02:25
||Let us move point '''B'''.

|-
||02:28
||Observe that the perpendicular line moves along with point '''B'''.
|-
|| 02:35
|| The perpendicular line and chord '''BC''' intersect at a point.

|-
||02:40
||Using the '''Intersect''' tool let’s mark the intersection point as '''D'''.
|-
|| 02:46
||Let’s measure the lengths '''BD''' and '''DC'''.
|-
||02:51
||Click on the '''Distance or Length ''' tool.

|-
||02:55
||Click on the points, '''B''' and '''D''' and then '''D''' and '''C'''.
|-
|| 03:01
||Notice that distances '''BD''' and '''DC''' are equal.

|-
||03:07
||It implies that '''D''' is midpoint of chord '''BC'''.

|-
||03:12
||Note that the perpendicular from the centre '''A''' to chord '''BC''' bisects it.

|-
|| 03:18
||Let us move all the labels using the '''Move''' tool to see them clearly.
|-
||03:28
|| Now let’s measure the angle '''CDA'''.

|-
||03:32
||Click on '''Angle''' tool and click the points '''C''', '''D''' and '''A'''.

|-
||03:39
||Angle '''CDA''' is 90 degrees.

|-
||03:42
||A line drawn from the centre to the midpoint of the chord is perpendicular to it.

|-
|| 03:48
|| Let us move point '''C''' and see how the distances change accordingly.

|-
||03:57
|| Pause the tutorial and do this assignment.

|-
|| 04:01
|| Open a new '''GeoGebra''' window.

|-
|| 04:04
|| Draw a circle.

|-
|| 04:06
|| Draw two chords of equal size to the circle.

|-
|| 04:10
|| Draw perpendicular lines from the centre to the chords.

|-
|| 04:15
|| Mark points of intersection.

|-
|| 04:18
|| Measure the perpendicular distances.

|-
|| 04:21
|| What do you observe?

|-
|| 04:23
|| The completed assignment should look like this.

|-
|| 04:27
|| Observe that, equal chords of a circle are equidistant from centre.

|-
|| 04:33
||Now let us go back to the circle.

|-
|| 04:36
|| Let us retain circle '''c''' and points '''A''', '''B''' and '''C'''.

|-
|| 04:43
|| Delete the rest of the objects.

|-
||04:46
||Go to the '''Algebra view'''.

|-
|| 04:49
|| Press the '''Ctrl''' key and select the objects for deletion.

|-
|| 04:54
|| Then press '''Delete''' key on the keyboard.

|-
||04:58
||Next let us prove a property with respect to an '''arc'''.

|-
|| 05:02
|| Inscribed angles '''BDC''' and '''BEC''' subtended by the same '''arc BC''' are equal.

|-
||05:10
||Let us next draw an '''arc'''.

|-
|| 05:13
|| Click on the '''Circular Arc''' tool.

|-
|| 05:16
|| Click on point '''A'''.

|-
|| 05:19
|| Then click on points '''B''' and '''C''' on the circumference.

|-
|| 05:24
|| An '''arc d''' is drawn.

|-
||05:27
||Let us change properties of '''arc d'''.

|-
|| 05:31
|| In the '''Algebra View''', right-click on object '''d'''.

|-
|| 05:35
|| Select '''Object Properties''' from the '''context menu'''.
|-
||05:39
||'''Properties''' window opens next to '''Graphics view'''.

|-
|| 05:43
|| Click on the '''Color''' tab and select green colour.
|-
|| 05:47
|| Let us change the style of filling of the '''arc d'''.

|-
|| 05:51
|| Select the '''Style''' tab and change the '''Filling''' to '''Hatching'''.
|-
|| 05:56
|| Close the '''Properties''' window.

|-
||05:59
|| Let us mark two points on the circumference of the circle.

|-
|| 06:04
|| Click on '''Point''' tool.

|-
|| 06:07
|| Mark point '''D''' above point '''B''' and point '''E''' above point '''C'''.

|-
|| 06:13
||Let us subtend two angles from '''arc BC''' to points '''D''' and '''E'''.

|-
||06:20
||Select the '''Segment''' tool and join the following points.

|-
|| 06:25
|| '''B,E''' '''E,C''' '''B,D''' and '''D,C'''.
|-
||06:33
|| Let’s measure the angles '''BDC''' and '''BEC'''.

|-
|| 06:38
|| Click on the '''Angle''' tool,

|-
|| 06:40
|| Click the segments that form the angle.

|-
|| 06:43
|| '''BD''' and '''DC''' and then click '''BE''' and '''EC'''.
|-
|| 06:51
||Observe that the angles '''BDC''' and '''BEC''' are equal.

|-
|| 06:57
|| This proves the property that angles formed using the same '''arc''' are equal.
|-
||07:04
||Let’s draw a sector '''ABC'''.

|-
|| 07:08
|| Click on '''Circular Sector'''tool.

|-
|| 07:11
|| Now click the points '''A''', '''B''', and '''C'''.

|-
|| 07:15
|| Sector '''ABC''' is drawn.
|-
||07:18
||Let’s measure the angle '''BAC''' using the '''Angle''' tool.

|-
|| 07:26
|| Observe that angle '''BAC''' is twice the angles '''BDC''' and '''BEC'''.

|-
||07:33
|| Using the '''Move''' tool let’s move point '''C''' to change the angles.

|-
|| 07:39
|| Notice the angles '''BEC''' and ''' BDC''' subtended by the '''arc d'''.

|-
|| 07:46
|| Angle '''BAC''' is always twice the angles subtended by the '''arc d'''.

|-
|| 07:52
|| Here angle at the centre is twice any inscribed angle subtended by the same '''arc'''.

|-
||08:00
||Next let us construct a pair of tangents to a circle.
|-
|| 08:05
|| Let us open a new '''GeoGebra''' window.
|-
|| 08:09
||Let us uncheck the '''Axes'''.
|-
||08:12
||Let's draw a circle using '''Circle: Center & Radius''' tool.

|-
|| 08:17
|| Click in the '''Graphics view''' to mark point '''A'''.
|-
||08:21
|| Type 3 for radius in the text box that opens.

|-
|| 08:26
|| Then click '''OK''' button.

|-
|| 08:29
|| A circle '''c''' with centre '''A''' and radius 3 centimetres is drawn.
|-
||08:35
||Now click on the '''Point''' tool.

|-
|| 08:38
|| Click to mark a point '''B''' outside the circle.
|-
||08:42
|| Using the '''Segment''' tool join points '''A''' and '''B''' to draw segment '''f'''.

|-
||08:49

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

|-
|| 08:54
|| Select the '''Perpendicular Bisector''' tool, click on points '''A''' and '''B'''.

|-
||09:01
||Segment '''f''' and perpendicular bisector intersect at a point.

|-
|| 09:07
|| Click on '''Intersect''' tool to mark the point of intersection as '''C'''.

|-
||09:12
|| Let's move point B.

|-
|| 09:15
|| Observe that perpendicular bisector and point '''C''' move along with point '''B'''.

|-
|| 09:22
|| This is because these objects are dependent on point ''' B'''.
|-
||09:27
|| Pause the tutorial and do this assignment.

|-
|| 09:31
|| Verify if point '''C''' is the midpoint of segment '''f'''.
|-
|| 09:36
|| Now let us draw another circle.
|-
||09:39
||Select the '''Compass''' tool.

|-
|| 09:42
|| Click on the points '''C''', '''B''' and '''C''' again to complete the figure.
|-
|| 09:48
|| Two circles intersect at two points.
|-
|| 09:52
|| Using the ''' Intersect''' tool, mark the points of intersection as '''D ''' and '''E'''.
|-
|| 10:00
|| Select the '''Segment''' tool.

|-
|| 10:03
|| Join the points '''B''', '''D''' and '''B''', '''E'''.
|-
|| 10:08
|| Segments '''h''' and ''' i''' are the tangents to circle '''c'''.
|-
|| 10:13
|| Let's explore some more properties of the tangents to the circle.
|-
||10:18
||Using the '''Segment''' tool and join the points '''A''', '''D''' and '''A''', '''E'''.
|-
||10:25
|| Let us show that triangles '''ABD''' and '''ABE''' are congruent.

|-
|| 10:32
|| Segment '''j''' is equal to segment '''k''', as they are radii of circle '''c'''.

|-
|| 10:39
|| In the '''Algebra view''' observe that segment '''j''' is equal to segment '''k'''.
|-
|| 10:47
|| Angle '''ABD''' is equal to angle ''' BEA''' ('''∠ADB''' = '''∠BEA''').

|-
|| 10:52
|| As they are angles on the semicircles of the circle '''d'''.

Let’s measure the angles.
|-
|| 11:01
|| Select the '''Angle''' tool.

|-
|| 11:04
|| Click the segments '''j''', '''h''' and '''i''', '''k''' to measure the angles.

|-
|| 11:11
|| Notice that are equal and 90 degrees.
|-
|| 11:16
||Segment '''f''' is the common side for both the triangles.

|-
|| 11:20
|| Therefore triangle '''ABD''' is congruent to triangle '''ABE''' by '''SAS''' rule of '''Congruence'''.
|-
|| 11:29
|| It implies that tangents '''BD''' and '''BE''' are equal.
|-
|| 11:35
|| From the '''Algebra view''', observe that segments '''h''' and '''i''' are equal.
|-
|| 11:41
|| Tangents are perpendicular to the radius of the circle at the point of contact.
|-
|| 11:47
||Let's move point '''B''' and see how the tangents move along with point '''B'''.

|-
|| 11:54
|| Tangents are drawn from point ''B''', so they are dependent on it.

|-
|| 12:00
|| Let’s now delete point '''B'''.

|-
|| 12:03
|| Right-click on point '''B''', from the context menu select '''Delete'''.

|-
|| 12:10
|| All the objects dependent on point '''B''' are deleted along with it.
|-
|| 12:16
|| We now have a circle '''c''' with centre '''A''' on the '''Graphics view'''.
|-
||12:21
||Select the ''' Point''' tool.

|-
|| 12:24
|| Mark points '''B''' and '''C''' on the circumference and '''D''' outside the circle.
|-
||12:30
||Select the '''Tangents''' tool.

|-
|| 12:33
|| Click on point '''D''' and then on the circumference.

|-
|| 12:37
|| Two Tangents are drawn to the circle '''c'''.
|-
||12:41
|| Tangents meet at two points on the circle.

|-
|| 12:45
|| Click on the '''Intersect''' tool and mark points of contact as '''E''' and '''F'''.
|-
|| 12:53
|| Let us draw a triangle.

|-
|| 12:56
|| Click on the '''Polygon''' tool.

|-
|| 12:59
|| Click on the points '''B''', '''C''', '''F''' and '''B''' again to complete the figure.
|-
||13:06
||In the figure segment '''b''' is the chord to the circle '''c'''.
|-
||13:11
||Angle '''FBC ''' is the inscribed angle by the chord '''CF''' to the circle '''c'''.
|-
|| 13:19
||Angle '''DFC''' is the angle between tangent and chord to circle '''c'''.
|-
||13:25
|| Let’s measure the angles.

|-
|| 13:28
|| Click on the '''Angle''' tool.

|-
|| 13:31
|| Click on the points '''F''', '''B''', '''C''' and '''D''', '''F''', '''C'''.
|-
|| 13:37
||Notice that angle '''DFC''' is equal to angle '''FBC '''.

|-
|| 13:46
|| Angle '''DFC''' is the angle between tangent and chord '''CF'''.

|-
|| 13:52
|| This angle is equal to inscribed angle '''FBC''' of the chord '''CF'''.

|-
||13:59
||Let's move point '''D'''.

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

|-
|| 14:08
|| Here all the objects are dependent on point '''D''' as the tangents are drawn from it.
|-
||14:16
|| Let us save this file now

|-
|| 14:19
|| Click on '''File ''' then ''' Save'''.

|-
|| 14:22
|| I will save the file on the ''' Desktop'''.

|-
|| 14:25
|| In the '''Save''' dialog box type the file name as '''Tangents'''.

Click on '''Save ''' button.

|-
|| 14:33
|| With this, we come to the end of the tutorial.

Let us summarise.
|-
||14:38
|| In this tutorial, we have learnt about the properties of, Chords, Arcs and sectors and Tangents

|-
||14:47
|| As an assignment.

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

|-
|| 14:52
|| Draw a circle.

|-
|| 14:54
|| Draw tangents from an external point.

|-
|| 14:57
|| Mark points of intersection of the tangents.

|-
||15:01
|| Join the centre of the circle to intersection points

|-
|| 15:05
|| Measure angle at the centre and measure angle between the tangents.

|-
|| 15:11
|| What is the sum of the two angles?

|-
|| 15:14
|| Join the centre and the external point.

|-
|| 15:17
|| Does the line segment bisect the angle at the centre?

|-
|| 15:22
|| The output of the assignment should look like this.

|-
|| 15:28
|| The video at the following link summarises the Spoken Tutorial project. Please download and watch it

|-
|| 15:36
|| We conduct workshops using Spoken Tutorials and give certificates. For more details, please contact us.

|-
|| 15:45
|| Please post your timed queries in this forum.
|-
||15:49
|| The '''Spoken Tutorial''' project is funded by the '''Ministry of Education '''Govt. of India.
|-
|| 15:55
|| This is Madhuri Ganapathi from, IIT Bombay signing off.

Thank you for watching.
|-
|}

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

PoojaMoolya: Created page with "{| border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to the Spoken tutorial on '''Properties of Circles''' in '''GeoGebra'''. |- ||00:07 || In this tutorial,..."