Geometry
Name ______________________
Period ___________
Date ___________
Symmetry and Transformations Using Geogebra
http://www.geogebra.org
This worksheet will guide you through the use of the Geogebra software program. We will use Geogebra
to discover many interesting ways to move or change a figure. This change is called a transformation.
Opening Geogebra
1.
2.
3.
4.
Open Internet Explorer and type http://www.geogebra.org into the address bar.
Click on the green “download” button.
Navigate to and click on the “Applet Start” button.
Wait until the Geogebra program opens (you will have to click “run” if a dialog box opens).
Reflections
The first transformation we will explore is called a reflection. A reflection uses a line like a mirror to
reflect an image. The line is called a line of reflection.
1. Navigate to the “Options” menu and click on “Advanced”. Click on the
icon and then
“grid”. Check the box that says “show grid” and close the options window.
2. Navigate to the fifth button from the left that looks like a triangle to open the polygon menu.
3. Click on the icon that looks like a triangle. This will let you draw a triangle.
4. Center your cursor over the point (1,2) on the grid and click, the point “A” should appear. Click
on the point (3,4) and (4,1) and again on (1,2) to create the Triangle ABC like the one shown
below.
5. We now want to create a line of reflection. Click on the
icon and choose “line through
two points”. Center your cursor over the point (0,1) on the grid and click, the point “D” with a
line through it should appear. Click on the point (1,1) to finish creating the horizontal line
through (0,1) and (1,1). Write the equation of the line. __________
6. To reflect triangle ABC over our line of reflection we need to access the transformation menu.
Navigate towards the
icon and click. Click on “reflect object about a line”.
We now need to choose our object to reflect, so click on triangle ABC (preimage).
Next, click on your line to reflect over it. Our new triangle (image) is
named A’B’C’ (say A prime B prime C prime).
Write the coordinates of A’, B’, and C’ below.
A’ = (
,
) B’ = (
,
) C’ = (
,
)
Questions to answer:
Does the triangle change size/shape?
If a figure’s size/shape does not change during a transformation, then it’s called an isometry.
Is a reflection an isometry?
What is the distance from A to the line of reflection?
What is the distance from A’ to the line of reflection?
Does this pattern continue for B and B’?
What do you notice about the coordinates of C and C’?
What happens to a point if it lies on a line of reflection?
7. To make things a bit easier, let’s erase triangle A’B’C’. Click on the “Edit” menu and then “undo”
to remove triangle A’B’C’.
8. Use the directions in step 5 to create a line through the points (2,4) and (2,0). What is the
equation of this line? ___________
9. Use the directions from step 6 to reflect triangle ABC over your new line of reflection.
Write the coordinates of the image below.
A’ = ( , ) B’ = ( , ) C’ = ( , )
Describe what happens to points A, B and C when they’re
reflected over this line. (Hint: See your answers from #6 above.)
10. On the following coordinate grid, plot the points A(-3,1) B(-1,3) and C(4,5). Connect these
points to make a triangle. Graph the line x = 1. Reflect triangle ABC over x = 1 and write the
coordinates of your image below.
A’ = (
,
) B’ = (
,
) C’ = (
,
)
11. Reflect the figure about the line y = x shown.
Symmetry
We can say an object has line symmetry if we can create a line of reflection that splits an object into two
congruent pieces. When we reflect an object over a line of symmetry it should look like it hasn’t moved.
Let’s explore this concept using triangles.
1. Under the file button in the upper left of the geogebra screen click open. Navigate to the G
drive, find your teacher and class period, then find the “threetriangles.ggb” file and click
“open”. You should have an isosceles, scalene and equilateral triangle on your screen.
2. Using the instructions from the previous exploration, create a line through points “G” and
“J”. Reflect triangle GHI over this line.
What happened when you reflected triangle GHI over the line? Does triangle GHI have
symmetry?
3. Under the edit menu, click “Undo” twice to remove the reflection and the line you created.
4. Using your ability to create lines and perform reflections, determine if triangle GHI has any
more lines of symmetry. In addition, determine if triangle ABC and DEF have any lines of
symmetry. Fill in the chart below with the number of lines of symmetry you found for each
triangle.
Triangle
Number of Lines of
Symmetry
5.
Triangle ABC
Triangle DEF
Triangle GHI
Use the coordinate planes below to draw the following:
a) Any polygon (with 4 or more sides) that has no line symmetry.
b) Any polygon (with 4 or more sides) that has one line of symmetry. Show the line of
symmetry.
c) Any polygon (with 4 or more sides) that has more than one line of symmetry. Show the
lines of symmetry.
a)
b)
c)