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Time Narration
00:01 Welcome to this tutorial on Types of Symmetry in GeoGebra.
00:06 In this tutorial we will learn about various types of symmetry like:



00:15 Rotation



00:22 To record this tutorial, I am using,

Ubuntu Linux OS version 14.04

GeoGebra version 5.0.438.0-d.

00:36 To follow this tutorial, you should be familiar with, the Geogebra interface.

If not, for relevant GeoGebra tutorials please visit our website.

00:49 Let us begin with the definition of symmetry.
00:53 A geometric figure is symmetric, if it can be divided into two or more identical parts and its parts can be arranged in an organized manner.
01:08 I have already opened the GeoGebra interface.
01:12 For this tutorial I will uncheck the axes.
01:16 To do so, right-click on the Graphics view.

The Graphics menu appears.

01:23 In this menu, uncheck the Axes check box.
01:27 For this tutorial, we will use all the tools available under the Reflect about Line tool.
01:35 Now we will define line symmetry.
01:38 A figure has line symmetry, if one half of the object is the mirror image of the other half.
01:46 The line over which the figure is reflected is called the line of symmetry.
01:52 To draw a vertical line AB, click on the Segment tool and then click on Graphics view.
02:00 Point A is drawn in the Graphics view.
02:04 Click again directly below point A to draw segment AB.

Note that it is labelled as f.

02:13 Select the Semicircle through 2 Points tool.
02:17 Click on the left-side of segment AB. Point C is drawn.
02:24 A gain click below C to complete the semicircle CD named as c.
02:30 This semicircle should be to the left of segment f.
02:35 Now let us reflect the semicircle about the segment f.
02:40 Click on the Reflect about Line tool. Click on the semicircle, then click on line f.
02:50 Semicircle c'(c prime) appears on the right side of segment f. It is a reflected image of semicircle c.
03:00 Let's change the object properties of c and c'(c prime).
03:05 Right-click on c and select Object Properties.
03:11 Preferences window opens.
03:14 In the left panel under Conic, c is already selected.
03:19 While holding the Ctrl key, click on c'(c prime).
03:23 In the Basic tab, click the Show Trace check box.
03:28 In the Color tab, I will choose the colour as pink.
03:33 You may choose any colour of your choice. Then close the Preferences window.
03:40 Using the Move tool, drag the semicircle c .
03:46 Observe that semicircle c'(c prime) moves as we move c.
03:52 c'(c prime) is the mirror image of c, with segment f as the mirror.
03:58 To erase the traces, drag the Graphics view.
04:03 Let us delete all the objects in the Graphics view.
04:07 Press Ctrl + A keys to select all the objects.
04:11 Then press the Delete key on the keyboard.
04:15 Now let us learn to reflect about a point.
04:19 Click on Segment tool.
04:22 Click within the Graphics view twice to draw a segment AB.
04:28 Select the Reflect about Point tool. Click on point A, then on point B.
04:38 A'(A prime)which is the reflected image of A, appears on the otherside of point B.
04:45 To view A' (A prime), drag the Graphics view if required.
04:50 To show that A'(A prime) is the image of A, we will measure the distances AB and A'(A prime)B.
04:58 Under Angle, click on the Distance or Length tool.
05:03 Click on point A, then on B.
05:08 Again click on A' (A prime) and then on B.
05:15 Notice that the distances AB and A'(A prime)B are equal.
05:20 Using Move tool, I will drag segment AB upwards.
05:27 Observe that A'(A prime) also moves along with AB.
05:32 Now we will learn to reflect a point about a circle.
05:36 Select the Circle with centre and radius tool. Click within the Graphics view.
05:43 The Circle with Centre and Radius text box appears.
05:48 In the text box type Radius as 2 and click on the OK button at the bottom.
05:56 A circle with centre C and radius 2 cm is drawn in the Graphics view.
06:02 Using Point tool, draw a point D outside the circle.
06:09 Select the Reflect about Circle tool. Click on point D and then click on circle c.
06:19 D'(D prime), which is the image of D, appears inside the circle.
06:24 Click on the Move tool and drag point D around the circle.
06:31 Observe that D'(D prime) also moves inside the circle mirroring D.
06:37 Drag point D inside the circle and see what happens. D and D'(D prime)exchange places.
06:47 Now let us learn about rotational symmetry.
06:51 An object has rotational symmetry, if it can be rotated about a fixed point without changing the overall shape.
07:02 Let us open a new GeoGebra window.
07:06 Click on File and then on New Window.
07:11 We will now rotate an object around a point. For this, I will draw a square.
07:18 Click on the Polygon tool.
07:21 Click within the Graphics view to draw point A. Similarly draw points B, C and D.
07:33 To complete the polygon click again on point A.
07:37 A quadrilateral ABCD named as q1 is drawn.
07:42 To convert q1 to a square, we have to adjust the lengths.
07:47 Click on the Move tool and drag the points A, B, C and D.
07:54 Notice the change in the lengths in the Algebra view. All the lengths have to be same.
08:01 We will now draw perpendicular bisectors to the square.
08:05 Click on Perpendicular Bisector tool.
08:08 Click on points A, B and B, C.
08:14 The two perpendicular bisectors intersect at a point.
08:18 Click on Intersect tool and click on point of intersection. Point E is the point of intersection.
08:28 Let us create an angle slider. Click on Slider tool and click in the Graphics view.
08:37 The Slider dialog box appears.
08:40 Select Angle radio button.
08:43 Alpha appears in the Name field.
08:47 Leave the default values of Min, Max and Increment as they are.

And click on the OK button at the bottom.

08:58 Alpha slider is created in the Graphics view.
09:02 Now click on the Rotate around Point tool. Click on the square q1 and then point E.
09:12 Rotate around Point text box appears with 45 degrees angle.
09:18 Below the text box we have, counter clockwise and clockwise radio buttons.
09:25 You can select any one of the radio buttons as per your choice.

I will select clockwise.

09:33 Delete 45 degrees from the Angle text box.
09:37 In the Angle text box, notice an alpha symbol on the rightside.
09:43 Click on it to show the table of symbols.
09:47 Select alpha from the table and click on the OK button at the bottom.
09:54 Observe that a new square q1' appears in the Graphics view.
10:00 This square q1' is rotated at angle alpha with respect to square q1.
10:07 Now drag the alpha slider between 0 degrees to 360 degrees.
10:13 As we drag, notice the rotation of q1' around the point E.
10:20 As an assignment,

Draw a hexagon and show its rotation symmetry.

10:28 Let us now delete all the objects.
10:31 Go to Edit menu and navigate to Select All.

Then select the Delete option.

10:41 Next we will move an object using a vector.
10:45 Let us define translational symmetry
10:49 An object has translational symmetry if, it can be moved without changing its overall shape.
10:58 Using the Polygon tool draw a triangle ABC named as t1.
11:08 To draw a vector, click on the Vector tool from the tool bar.
11:13 Click on point D and then on point E.
11:19 The vector is represented by u.
11:23 Select Translate by Vector tool. Click on the triangle t1 and then on the vector u.
11:33 Here t1' is the translated image of t1.
11:38 The distance between t1 and t1' is exactly same as the length of vector u.
11:45 Using the Move tool, drag point E of the vector u.

Observe that the image triangle t1' translates along with vector u.

11:59 As an assignment,

Draw a vector.

12:04 Translate a point using Translate by Vector tool.
12:08 Measure the distance between the original point and the translated point.
12:13 Let us define scale symmetry.
12:16 An object has scale symmetry if, it does not change shape when it is expanded or contracted.
12:25 Let us open a new Geogebra window. Click on File and select New Window.
12:34 Now let us learn how to dilate an object.
12:38 Click on the Circle with centre and radius tool. Then click on the Graphics view.
12:45 Type radius as 1 in the Circle with Centre and Radius text box.
12:50 Click on OK button at the bottom.
12:53 Using Point tool draw a point B outside the circle.
12:59 Select the Dilate from Point tool.
13:02 Click on the circumference of the unit circle then click on point B.
13:09 The Dilate from Point text box appears. Type the Factor as 2 and click on the OK button at the bottom.
13:20 A dilated circle with double the radius appears in the Graphics view.
13:26 As an assignment,

Draw a pentagon and a hexagon on the same window.

13:32 Dilate the pentagon by a factor of 0.5

Dilate the hexagon by a factor of 3.

13:40 Let us summarise what we have learnt.
13:44 In this tutorial we have learnt about

Symmetry and various types of symmetry



13:56 Rotation



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