ApplicationsofGeoGebra/C3/3DGeometry/English
Visual Cue  Narration 
Slide Number 1
Title Slide 
Welcome to this tutorial on 3D Geometry. 
Slide Number 2
Learning Objectives 
In this tutorial, we will learn how to use GeoGebra to view:
And construct different structures in 3D space Solids of rotation of polynomial functions Trigonometric functions in 3D space 
Slide Number 3
System Requirement 
Here I am using:
Ubuntu Linux OS version 16.04 GeoGebra 5.0.481.0d 
Slide Number 4
Prerequisites www.spokentutorial.org 
To follow this tutorial, you should be familiar with:
GeoGebra interface Geometry For relevant tutorials, please visit our website. 
Slide Number 5
Rectangular Coordinate System [[Image:]] 
This image shows the rectangular coordinate system.
It is made up of mutually perpendicular axes and planes formed by them. The axes are x (in red), y (in green) and z (in blue). All points in 3D space are denoted by their x y z coordinates. The point of intersection of the three axes is the origin O 0 comma 0 comma 0. The gray rectangle in the image depicts the XY plane. The planes divide space into 8 octants. Point A is in the XOYZ octant and has the coordinates 4 comma 4 comma 2. 
Show the GeoGebra window.  Let us draw a 3D pyramid in GeoGebra.
I have already opened a new window in GeoGebra. This time, we work with Algebra, 2D Graphics and 3D Graphics views. 
Under View, select 3D Graphics.  Under View, select 3D Graphics. 
Click in 2D Graphics View to draw in 2D.  Click in 2D Graphics View to draw in 2D. 
Drag the boundary to see 2D Graphics properly.  Drag the boundary to see 2D Graphics properly. 
Click in 2D Graphics.  Click in 2D Graphics. 
In 2D Graphics view, click on Polygon tool and click on origin (0,0).
Point to A. 
In 2D Graphics view, click on the Polygon tool and click on origin 0 comma 0.
This creates point A at the origin. 
Click on (2,0) to create point B.  Then click on 2 comma 0 to create point B. 
Click on (2,2) for point C >>
on (0,2) to draw point D. 
Click on 2 comma 2 for C and on 0 comma 2 to draw D. 
Click again on point A.  Finally, click again on A. 
Point to quadrilateral q1 in 2D >> 3D Graphics views.  Note that a quadrilateral q1 is seen in 2D and 3D Graphics views.

Click on Move tool.  Click on the Move tool. 
Click in 2D Graphics >> drag the background.  Click in 2D Graphics and drag the background. 
Drag the boundary to see 3D Graphics properly.  Drag the boundary to see 3D Graphics properly. 
Click in 3D Graphics and under Pyramid,
on the Extrude to Pyramid or Cone tool. Click on the square in 3D Graphics view. 
Click in 3D Graphics and under Pyramid, on the Extrude to Pyramid or Cone tool.
In 3D Graphics view, click on the square. 
An Altitude text box opens up, type 3 and click OK.  In the Altitude textbox that opens, type 3 and click OK. 
Point to pyramid e in 3D Graphics view.
Point to base. Point to E (1,1,3). Show altitude. 
A pyramid e appears in 3D Graphics view.
Its base is the quadrilateral q1. Its apex is E 1 comma 1 comma 3. Its altitude or height is 3 units. 
Slide Number 6
Rotation of a Polynomial Let us rotate f(x)= 2x^{4}x^{3}+3x^{2} Part in second quadrant (XY plane) about xaxis 
Rotation of a Polynomial
Let us rotate f of x equals minus 2 x raised to 4 minus x cubed plus 3 x squared. We will rotate the part that lies in the second quadrant, in XY plane, about the xaxis. 
Show the GeoGebra window.  I have already opened a new window in GeoGebra.
We will initially work with Algebra and 2D Graphics views and open 3D Graphics view later. 
In input bar, type the following line
>> press Enter. f(x) = 2 x^4 x^3+3 x^2 
In the input bar, type the following line.
To type the caret symbol, hold Shift key down and press 6. Spaces here denote multiplication. Press Enter. 
Under Perpendicular Line, click on Parallel line
>> on the yaxis. 
Under Perpendicular Line, click on Parallel line and on the yaxis. 
Keeping the cursor on the xaxis.  Keep the cursor on the xaxis. 
Drag it along until you reach the intersection of f and xaxis.
Point to the label function f, xaxis that appears. 
Drag it along until you see function f, xaxis at the intersection of f and xaxis. 
Click on this intersection point.  Click on this intersection point. 
Point to A.  Point A appears. 
Click on Slider >> in Graphics view.  Click on Slider and in Graphics view. 
A Slider dialog box opens.  A Slider dialogbox opens. 
Leave a as the Name.  Leave a as the Name. 
Change Min value to 1.5, Max value to 0
Increment to 0.05. 
Change Min value to minus 1.5, Max value to 0 and Increment to 0.05. 
Click OK.  Click OK. 
Point to slider a.
Point to the part of function, xaxis in the second quadrant. 
This creates slider a, which changes the value of a from 1.5 to 0.
It will focus on the part of the graph in the second quadrant. 
In the input bar, type (a,f(a)).
Press Enter. 
In the input bar, type the following in parentheses.
a comma f a in parentheses. Press Enter. 
Point to B.  This creates point B whose x coordinate is the value of a.
Its ycoordinate lies along the curve described by the function f between x equals 1.5 and 0. 
Rightclick on slider a >> check Animation On.  Rightclick on slider a and check Animation On. 
Point to B and slider a.  Point B travels along function f as a changes. 
Rightclick on slider a >> uncheck Animation On.  Rightclick on slider a and uncheck Animation On. 
In the input bar, type (a,0) >> press Enter.  In the input bar, type a comma 0 in parentheses and press Enter. 
Point to C and slider a.  This creates point C.
As its x coordinate a changes, C moves below point B along the xaxis. 
Under Line, click on Segment
>> click on B and C to join them. 
Under Line, click on Segment and click on B and C to join them. 
Click on Move Graphics View >> drag background to the left.  Click on Move Graphics View and drag the background to the left. 
Click on View
>> check 3D Graphics to see the 3D Graphics view. 
Click on View and check 3D Graphics to see the 3D Graphics view. 
Point to 2D Graphics and 3D Graphics views.  Note that what is drawn in 2D Graphics appears in the XY plane, in 3D Graphics. 
Click in 3D Graphics view >> on Rotate 3D Graphics View.
Rotate the 3D Graphics view to see the curve properly. 
Click in 3D Graphics view and on Rotate 3D Graphics View.
Rotate 3D Graphics to see the curve properly. 
Place cursor on yaxis (in green).
Click to see an arrow aligned with the yaxis. 
Place the cursor on the yaxis (in green).
Click to see an arrow aligned with the yaxis. 
Drag along the y axis to see the curve properly.  Drag to pull the yaxis in or outwards to see the curve. 
Type Circle[C,f(a),xAxis] in the input bar
>> press Enter. 
In the input bar, type the following line. 
Point to circle c with center at point C.  This creates circle c with center at point C.
Its radius is equal to f of a corresponding to the value of a on slider a. Its rotation is around the xaxis. Press Enter. 
Rightclick on circle c in Algebra view
>> check Trace On option. 
In Algebra view, rightclick on circle c and check Trace On option. 
Rightclick on slider a >> select Animation On option.  Rightclick on slider a and select Animation On option. 
Point to the solid traced as a changes.  Observe the solid traced as a changes. 
Point to 2D >> 3D Graphics views.  Watch both 2D and 3D Graphics views. 
Point to Segment BC, xaxis and function f.  Segment BC moves between the xaxis and function f. 
Point to the part of f in the second quadrant.  The part of function f that is in the second quadrant in 2D, rotates around the xaxis. 
Drag 3D Graphics to see it from another angle.  Drag 3D Graphics to see it from another angle. 
Show the GeoGebra window.  Finally, let us look at trigonometric functions in 3D.
I have already opened a new window in GeoGebra. 
Click on 3D Graphics tool under View.  Under View, click on 3D Graphics. 
Drag the boundary to see 2D Graphics properly.  Drag the boundary to see 2D Graphics properly. 
Click in 2D Graphics
>> Slider tool and in Graphics view. 
Click in 2D Graphics, then on the Slider tool and in Graphics view. 
Point to the slider dialog box and Number radio button selection.
Type t in the Name field. 
A slider dialogbox opens.
By default, the Number radiobutton is selected. In the Name field, type t. 
Set Min to 6, Max to 16 and increment of 0.1.
Click OK. 
Set Min to minus 6, Max to 16 and increment of 0.1.
Click OK. 
Point to slider t.  This creates a slider t which will change t from minus 6 to 16. 
In the input bar, type f(t)=cos(t)
>> press Enter. 
In the input bar, type f t in parentheses equals cos t in parentheses and press Enter. 
Click in 2D Graphics.  Click in 2D Graphics. 
Under Move Graphics View, click on Zoom Out
>> click in 2D Graphics. 
Under Move Graphics View, click on Zoom Out and click in 2D Graphics. 
Click on Move Graphics View >> drag the background.  Click on Move Graphics View and drag the background. 
Point to the cosine function of f(t) in 2D and 3D Graphics views.  You can see the graph of the cosine function of f of t, in 2D and 3D Graphics views. 
Type g(t)=sin(t) in the input bar >> press Enter.  Similarly, in the input bar, type g t in parentheses equals sin t in parentheses.
Press Enter. 
Point to sine function graph (g(t)).  Sine function graph of g of t appears. 
Type h(t)=t/4 in the input bar >> press Enter.  In the input bar, type h t in parentheses equals t divided by 4 and press Enter. 
Point to line (h(t)).  Line h of t is of the form y equals mx where slope m is 1 divided by 4. 
Click in 3D Graphics view.  Click in 3D Graphics view. 
Click on Point tool >> click in the gray area in 3D Graphics view.
Point to point A. 
Click on the Point tool and click in the gray area in 3D Graphics view.
This creates point A. 
Drag the boundary to see its coordinates properly.  Drag the boundary to see its coordinates properly. 
Double click on point A in Algebra view.
Change the coordinates to (f(t),g(t),h(t)). Press Enter. 
In Algebra view, doubleclick on A.
Change the coordinates to the following. Press Enter. 
Point to A.  The x coordinate of A is cos t.
The ycoordinate is sin t and t divided by 4 is its z coordinate. 
Rightclick on slider t >> click on Object Properties.  Rightclick on slider t and click on Object Properties. 
A Preferences dialogbox opens.
Click on Slider tab. Under Animation, for Repeat, choose option Increasing from dropdown menu. 
A Preferences dialogbox opens.
Click on Slider tab. Under Animation, for Repeat, choose option “Increasing” from the dropdown menu. 
Close the Preferences dialog box.  Close the Preferences dialog box. 
Rightclick on point A in Algebra view
>> select Trace On. 
In Algebra view, rightclick on A and select Trace On. 
Rightclick on slider t >> check Animation On.  Rightclick on slider t and check Animation On. 
Point to point A and the helix in 3D Graphics view.
Point to point A’s coordinates in Algebra view. 
Point A traces a helix in 3D space with coordinates mentioned earlier. 
Click in Rotate 3D Graphic View >> rotate the background.
Rotate 3D Graphics view. 
Click in Rotate 3D Graphic View and rotate the background.
Rotate 3D Graphics view so you are looking down the zaxis at the XY plane. 
Point to the traces of point A (cos(t), sin(t)).  Note that the traces of A are the circumference of a unit circle.
Point A moves along the circle as angle t changes. In 2D, its coordinates are cos t comma sin t. 
Let us summarize.  
Slide Number 7
Summary 
In this tutorial, we have learnt how to use GeoGebra to view:
And construct different structures in 3D space Solids of rotation of polynomial functions Trigonometric functions in 3D space 
Slide Number 8
Assignment Construct a prism and a cylinder. Draw lines to pierce the structures and find their intersection points. Graph the polynomial, f(x)=x^{5}7x^{4}+9x^{3}+23x^{2}50x+24. Show the solid formed due to rotation of peak in first quadrant in XY plane. 
As an assignment:
Construct a prism and a cylinder anywhere in 3D space. Draw lines to pierce the structures and find their intersection points. Graph the given polynomial. Show the solid formed due to rotation of the peak, in the first quadrant, in the XY plane. 
Slide Number 9
Assignment You tried to fly a kite off a cliff. The kite got dumped into the lake below You gave out 325 feet of string The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees How high is the cliff? 
As another assignment,
You tried to fly a kite off a cliff. The kite got dumped into the lake below. You gave out 325 feet of string. The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees. How high is the cliff? 
Slide Number 10
About Spoken Tutorial Project 
The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it. 
Slide Number 11
Spoken Tutorial workshops 
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Slide Number 12
Forum for specific questions: Do you have questions in THIS Spoken Tutorial? Please visit this site Choose the minute and second where you have the question Explain your question briefly Someone from our team will answer them 
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Slide Number 13
Acknowledgement 
Spoken Tutorial project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link. 
This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. 