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

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 theAlgebra 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 900 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 Sectortool. 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 900 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.

Contributors and Content Editors

Madhurig