Learning geogebra

= 6.Learning geometry with GeoGebra =

Chapter objectives
In this chapter you will learn


 * How you can use the computer to learn 	mathematics.
 * About GeoGebra a mathematical software 	application that helps you learn Geometry, Algebra and Statistics.
 * How to use GeoGebra.
 * Connect GeoGebra with the geometry taught in 	the maths class to better understand some of the concepts and 	properties.

How can we use ICTs for learning mathematics?
Nowadays we see computers everywhere and we know that it has become an important part of our daily lives. So far we have learnt how to use the computer for basic document writing also called text processing. We have used the Internet to read information. We might also have watched some videos, listened to music, seen some CDs about our class lessons. You must have played some games on the computer as well! Have you seen or used a calculator? A calculator is a device to do arithmetic calculations. Now calculators are available in computers and mobile phones. Now you can also learn other areas of mathematics such as geometry, algebra and statistics using software applications like GeoGebra.

It helps you visualise geometric concepts and explore properties of different types of geometric figures like lines, triangles and rays.

What is GeoGebra ?
Can you guess why the creators of this tool named it GeoGebra. Geo is an abbreviation for Geometry and Gebra for Algebra. Did you know that you can represent all geometric figures as algebra equations? GeoGebra helps you see both the geometric figures and its algebra representation at the same time, hence the name GeoGebra.



In this chapter you will learn to understand Geometry concepts using GeoGebra.

In GeoGebra you can animate the geometric figure you have drawn and dynamically  see how some values like length, area, perimeter of a figure changes, see the same figure in different ways.



Introduction, the GeoGebra window
The GeoGebra screen is divided in several sections to represent mathematical objects in different

ways. The names of the different parts are shown below.

''Main Screen - GeoGebra''

'''Menu Bar: It is a '''typical windows command menu bar. '''File, Edit, View '''etc... are called the menu-items.

Tool Bar:  It has all the tools (compass box) to use in the graphic view.

Active Tool View: It tells you which tool is active to use on the graphic view and how to use it.

Graphic View: It is used to draw the geometric figures (also called objects in GeoGebra). This window can never be closed.

Algebra View: It shows the algebraic expressions. This window can be closed if you are working only on geometry.

Input Bar: This is used to enter more complex mathematical expressions that may not be available on the Tool Bar.

Commands: It has to be used along with the Input Bar, to select from a list of available commands.

The Toolbar
The tool bar is GeoGebra's compass box

Each tool has many related tools under it. To see all the related tools, click on the arrow at the bottom right hand corner of each tool as shown below.

Basic use of tools 

Look of screen
 * [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m2e9355f9.png]]Activate 	a tool by clicking on the button showing the corresponding icon.
 * Open a toolbox by clicking on the lower part of a 	button and select another tool from this toolbox.

In GeoGebra only the graphic view cannot be closed. You can change the screen to be best fit for working with your needs: Press View on the menu Item and uncheck or check Algebra View, Axes and Grid based on your needs.

GeoGebra Exercises

 * 1) Drawing 	points, line segment and rays

To learn how to use GeoGebra we will use the most commonly used tools when drawing geometrical shapes.

Can you describe in your own words the difference between a segment, line and ray? Also see the algebra view and observe the equations of the line b and ray c. The line segment a is represented in the algebra view as a = 2.83, where 2.83 is the length of the segment.
 * 1) Select 	Point Tool, and click anywhere on the drawing point to plot six 	points A, B, C, D, E, F.
 * 2) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_6802121.png]]Select 	Segment between two points tool, click on point A and then 	point B.
 * 3) Select 	Line through two points tool, 	click on point C and then point D.
 * 4) Select 	Ray through two points 	tool, click on point E and the point F.

Now use the Move Tool move points A, B and C. What do you observe? Describe it.
 * 1) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_1eb2590.png]]Drawing 	a parallel line
 * 2) Select 		Point Tool and click anywhere on the drawing point to plot three 		points A,B, C.
 * 3) Select 		Line through two points tool, 		click on point A and then point B.
 * 4) Select 		Parallel Line tool, 		click on point C first. Then click on line AB.

Next use the Move Graphics view tool and move the drawing pad. Do the two lines ever meet?

Try making a pentagon and hexagon on your own.
 * 1) Drawing 	 polygons
 * 2) Select Point tool and plot three points A B and C to 		represent the vertices of a triangle.
 * 3) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m54b722a1.png]]To 		draw a three-sided polygon – triangle, select Polygon 		tool click on point A, then B and C and again click on point A. OR

In this exercise you are going to draw a right-angled triangle where the base is 5 units and the hypotenuse is 8 units.
 * 1) Rotate 	a ray
 * 2) Draw 		line segment AB of any length (Segment between two points tool).
 * 3) Select the Ray Through two points tool, click on point A, 		then select another point C on the drawing pad as shown in the 		figure.
 * 4) Select the Angle tool, as seen in the figure and click on 		points B, then A and finally C. You will see an angle measure. 		Click on the Move tool and move point C. Observe the change of 		angle.
 * 5) Observe the direction (clockwise, anticlockwise) in which you move 		the ray? In which direction does the angle increase and which 		direction does it decrease?
 * 6) Draw 	triangles
 * 1) Draw 	triangles

All buttons on the tool bar hide many related tools. You choose the tool you want by pressing the small red arrow in the lower right corner of the button. Choose from the list that shows up.


 * 1) Start your drawing by using the 	tool Segment with Given Length 	from Point.


 * 1) Continue by drawing the right 	angle. Do this by drawing a perpendicular line through point A. 	 Choose the perpendicular line tool, click on point A first and then 	on the line.


 * 1) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_72a9f629.png]]To 	mark the third corner of the triangle you use one of the circle 	tools, Circle with Centre and 	Radius.
 * 2) Click on the point B and 	fill in the length of the hypotenuse as radius.


 * 1) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m5680d7e4.png]]Choose 	the tool Intersect Two Objects, 	click on the circle and the perpendicular line. The point in the 	intersection is the third corner of the triangle.




 * 1) Draw the triangle by choosing 	the Polygon tool. You need to click all the corners 	and then click again on the first corner to complete the triangle.


 * 1) The perpendicular line and the 	circle, even the points do not need to be visible or seen now, you 	only want to show the triangle. Hide an object by right-clicking 	the object and uncheck Show 	Object by 	clicking on it.




 * 1) The lengths of the sides in the 	triangle can be shown. Right-click one of the sides and choose 	Object Properties in 	the menu which shows up. Check the Show 	Label field and choose Value 	from the drop down list.
 * 1) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_74ac619e.png]]To 	show the size of the angles use the Angle tool. Click on each 	vertex of the triangle. The order in which you click the 	vertices must be in the clock wise direction. In this figure click 	in this order BAC, CBA, and ACB.

To save GeoGebra files
 * 1) The area of the triangle can also be shown, 	use the Area tool as seen in the figure above. Click on Area 	tool and then click on the polygon.
 * 2) Change the shape of the triangle by moving 	the points you are able to move (use the Move tool).


 * 1) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m30e17108.png]][[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m6e1d0aba.png]]To 	save your GeoGebra file, select menu item File &gt; Save As.
 * 2) Select the Document folder.
 * 3) Type the filename Right Angled Triangle.
 * 4) The Files of Type box will automatically have 	Geogebra Files (.ggb).
 * 5) Click Save. Your file will be saved as Right 	Angled Triangle.ggb.

To open GeoGebra files


 * 1) Open the GeoGebra applications.
 * 2) Select menu item 	File &gt; Open.
 * 3) Select the folder 	and the file.
 * 4) Click Open.

'''Additionally you can try the following exercise.'''


 * 1) Follow the following steps to construct a 	rectangle ABCD like the figure. Use the Move tool 	and move the vertices of the rectangle. If you have constructed the 	rectangle correctly, when you move the vertices the figure will 	always be a rectangle. 	Seeing this construction can you write down the properties of a 	rectangle?   Steps:
 * 2) Draw a line 		segment AB of any length (Segment between two points tool).
 * 3) Draw a 		perpendicular line at point A perpendicular to line segment AB. For 		this in GeoGebra : Select the Perpendicular line tool. Click 		on point A then select line segment AB.   [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_m7091d368.png]]
 * 4) Similarly draw 		another perpendicular line at point B to line segment AB. How will 		you do this in GeoGebra?
 * 5) Now select the 		Parallel Line tool and draw a parallel line to AB at 		any new point C.
 * 6) Mark the 		intersection points of the parallel and perpendicular lines as D 		and E using the Intersect two objects tool.
 * 7) ADEB the 		rectangle you constructed.
 * 8) To measure the 		sides, use the Distance or Length tool and click on points 		AD, then DE, then EB and finally AB.
 * 9) Select the move 		tool and move points A or B. What do you observe, write down your 		observations.
 * 10) Now use the 		polygon tool and complete your rectangle. Also hide the 		construction. How will you do this?

Chapter summary
In this chapter you have learnt about:


 * 1) GeoGebra 	is a mathematical tool that helps you construct geometrical figures 	and see the geometric and algebraic representations at the same 	time. It is dynamic (where you can change the values of some of the 	properties like length, angle) and you can animate 	your geometric figures to understand properties of the figures you 	have drawn.
 * 2) You learnt how to 	use GeoGebra to construct different geometrical figures and 	understand some mathematical concepts.

Exercises

 * 1) Verifying a Theorem: The 	sum of the interior angles of a triangle are 180 degrees.
 * 2) Draw 		three points A, B, C (New 		Point tool)
 * 3) Draw 		the triangle with vertices A,B and C (Polygon 		tool)
 * 4) Select 		the Angle tool 		to measure each of the interior angles of the triangle. Now verify 		that the sum of all the interior angles equals to 180 degrees.
 * 5) Select one of the vertices of the triangle (A,B or C) 		and move the points (Move tool) to change the shape of the 		triangles.
 * 1) Exploring 	the rule: Similar triangles are proportional : Draw a triangle. 	Place a point on one of the sides and draw a line through this point 	parallel to one of the sides in the triangle. Parallel line: Use the 	Parallel line tool. You now have two triangles like the 	figure below.   [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_76cbafdc.png]]
 * 1) Explain why triangle 	ABC is similar to triangle DBE. Measure the lengths of the sides in 	both triangles.
 * 2) Check if the rule stating that 	the ratio between the lengths of corresponding sides in similar 	triangles is the same applies.
 * 3) Measuring lengths: Use the 	Distance or Length tool.
 * 4) Make the Algebra View visible: 	View on the Menu Item.
 * 5) Show names of points: 	Right‐click on one of the points, choose properties, in the list 	on the left side click on points and check Show Label.
 * 6) Calculate the ratio: In the 	Input bar write: r1=Distance[A,B]/Distance[A,D]. AB and AD are 	corresponding sides in the two triangles. Repeat the command for the 	other two pairs of corresponding sides.
 * 7) Change the shape of the 	triangle by moving the points you are able to move (use the Move 	tool). Are the triangles still similar? What happens to the ratios?


 * 1) Construct 	a square in GeoGebra. You are only given a line segment AB of any 	length. How will you construct the square? The steps are given 	below
 * 2) Draw 		a line segment AB of any length using Segment between Two Points 		Tool.
 * 3) Draw 		two perpendicular lines at points A and B respectively and 		perpendicular to the line 		segment AB using Perpendicular Line tool.
 * 4) Draw 		a circle of radius = length of line segment AB using the circle 		with centre through point tool. 		Make point A the centre of the circle and point B as the point on 		the circle.
 * 5) Use 		Intersection two objects and 		click on the point where the circle intersects the perpendicular 		line at A. This will be point C.
 * 6) Draw 		a parallel line by selecting point C and line segment AB using the 		parallel line tool.
 * 7) Use 		Intersection two objects and 		click on the point where the parallel line intersects the 		perpendicular line at B. This will be point D.
 * 8) ABCD is your square.
 * 9) Measure the sides using the 		Distance or Length tool. 		What do you observe? What are the properties of a square?
 * 10) [[Image:ICT%20Phase%203%20%20-%20Resource%20Book%208th%20Standard%20ENGLISH%20-%2070%20Pages_html_3d47a81a.jpg|50px]]Move 		the points A or B using the Move tool. Is 		ABCD still a square?

Additional resources

 * 1) To learn more GeoGebra seethe 	website []
 * 2) To download GeoGebra files 	(.ggb) created by your mathematics teachers go to 	[] 	 Select Maths Tab, then Computer Tools Tab.
 * 3) [] 	 To see video tutorials of GeoGebra
 * 4) To learn about how to 	participate in mathematics olympiads please see the website 	[]
 * 5) Here are a list of mathematics 	websites that you can try out puzzles
 * 6) [] []