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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,


00:12 Arcs and sectors and


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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