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

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 theAlgebra 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 Sectortool.
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.

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