GeoGebra-5.04/C3/Properties-of-Circles/English-timed
From Script | Spoken-Tutorial
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. |