GeoGebra Tutorial 1 – Quadrilaterals & Midpoints
http://math4allages.wordpress.com/2009/11/04/geogebratutorial1/
Problem: Investigate what happens if you connect the consecutive midpoints of a
quadrilateral.
In this tutorial, we will use GeoGebra to explore what happens if we connect the
consecutive midpoints of a quadrilateral. We will learn to use the following tools:
Move, Midpoint or Center, Segment Between two points and Pol
Polygon.
ygon. You can
follow this tutorial step by step by opening the GeoGebra window.
1.) We will not need the Algebra window and the Coordinate axes so we
will hide them. To hide the Coordinate axes, click the View menu on the
menu bar, and then click Axes. To hide the Algebra window,
click View then click Algebra window.
2.) Click the New Point button
button,, and then click four distinct places on
drawing pad forming the vertices of a quadrilateral.
3.) If the labels of the points are not displayed, click the Move button,
button
right click each point and click Show label from the context menu.
(The context menu is the pop
pop-up
up menu that appears when you right click an
object.)
4.) Select the Segment between two points button and click point A and
click point B two distinct points to construct segment AB.. To
construct BC,, click point B and another location to create segment BC.
Repeat until ABCD is formed. After step 4, yourr drawing should look like
Figure 1 below.
Figure 1 - Quadrilateral ABCD
5.) Click the Move button and move the vertices of the quadrilateral.
What do you observe?
6.) To determine the midpoint of each side of the quadrilateral, click the
inverted triangle at the bottom right of the New Point button and choose
Midpoint or Center tool from the list. Click the segments in the following
order: AB, BC, CD and AD.
7.) Select the Move button, right click the midpoints and click Show label
from
m the context menu. After step 7, your drawing should look like Figure
2.
Figure 2 - Quadrilateral ABCD with Midpoints E, F, G and H.
8.) Move the vertices of the quadrilateral. What do you observe?
9.) To have a better view, let us connect the midpoints of the quadrilateral
using the Polygon tool. To do this, click the Polygon tool and click the
points in the following order: Point E, point F, point G,, point H and then
point E again to close the polygon.
10. Move the vertices of the qua
quadrilateral.
drilateral. What do you observe about the
figure?
11. What conjecture can you make based on your observation?
GeoGebra Tutorial 2 – Constructing an Equilateral Triangle
http://math4allages.wordpress.com/2009/11/09/geogebra
http://math4allages.wordpress.com/2009/11/09/geogebra-tutorial-2-constructing
constructing-an-equilateraltrianlge/
Problem: How will you draw an equilate
equilateral
ral triangle without using the
Regular polygon tool?
In this tutorial, we will mimic compass and straight edge construction using
the circle tool. The idea is to use the intersections of two circles and the two
centers to form as triangles as shown below.
Figure 1 - An equilateral triangle formed by radii of two circles
In constructing our triangle, we are going to learn how to use the circle tool,
and how to display interior angle measure of a polygon and length of
segments. Note that we can also draw a regular polygon of any side by using
the Regular polygon tool.
1.) We will not need the Algebra window and the Coordinate axes so we
will hide them. To hide the Coordinate axes, click the View menu on the
menu bar, and then click Axes. To hide the Algebra window,,
click View then click Algebra window.
2.) Click the Segment between two points tool, and click two distinct
places on the drawing pad to construct segment AB.
3.) If the labels of the points are not displayed, click the Move button,
right click each point and click Show label from the context menu.
(The context menu is the pop-up
up menu that appears when you right click
an object.)
4.) To construct a circle with center A passing through B,, click the Circle
with Center through
ugh Point tool, click point A,, then click point B. After
step 4, your drawing should look like the one shown in Figure 2.
2
Figure 2 - Circle with center A and passing through B.
5.) To construct another circle with center B passing through A, with
the Circle with Center though Point still active, click point B and then
click point A.
6.) Next, we have to intersect the circles. To intersect the two circles, click
the inverted triangle on the New Point tool, select Intersect Two
Objects, then clickk the circumference of both circles. Notice that two
points will appear in their intersections. After step 6, drawing should look
like the one shown in Figure 3.
Figure 3 - Circles with radius AB and intersection C and D.
7.) We only need three points, points A, B and C,, to form an equilateral
triangle, so we will hide the two circles, segment AB and point D. To do
this, right click each object and click the Show Object option to uncheck
it. In hiding segment AB, be sure that
hat you do not click points A or B.
8.) With only three points remaining on the drawing pad, click
the Polygon tool and click the points in the following order: point A,
point B, point C and then point A to close the polygon. After step 8, your
drawing should look like the figure below. Note that in other versions, or
depending on the setting, segment labels may appear.
Figure 4 - Triangle formed from radii of two circles
9.) Using the Move button, move the vertices of the triangle. What do you
observe? Notice that you can move vertex A and B but you can never
move vertex B.. This is because vertex B is a dependent object. Recall that
vertex C is the intersection of two circles and thus depends on the length
of segment AB.
10.) You have prob
probably observed that it seems that ABC is an equilateral
triangle. In fact, it is. To verify, we can display the interior angles of the
triangle. To do this, click the Angle tool,, then click the interior of the
triangle.
11.) What do you observe? Move the
he vertices of the triangle. Is your
observation still the same?
12.) You can also verify the length of the sides using the Properties
window. To do this, right click one of the sides of the triangle, click
Object Properties from the context menu.
13.) In the Object Properties window, select the Basic tab. Be sure that
the Show label check box is checked and choose Value from the Show
label drop down list box.
14.) Select the other sides of the triangle in the Object list located at the
left side of the Object Properties window and change the labels
to Value, then close the window when you are done.
15.) Prove that the construction above always results to an equilateral
triangle.
GeoGebra Tutorial 3 – Constructing a Square
http://mathandmultimedia.com/2009/11/10/tutorial
http://mathandmultimedia.com/2009/11/10/tutorial-3-constructing-a-square/
square/
Problem: How will you draw an equilateral triangle without using the
Regular polygon tool?
In this tutorial, like Tutorial 2, we will mimic compass and straightedge
construction using the circle tool, the parallel line tool, the perpendicular tool
to construct a square instead of using the Regular polygon tool. We will also
reinforce the use of the angle tool, this time, learn how to use it to measure
angle using three points.
Figure 1 - Square formed from radii of a circle
The idea of our construction is to construct a circle with radius AB and
construct lines parallel and perpendicular to it to form our square.
START
1.) Open the GeoGebra w
window. We will not need the Algebra
window and the Coordinate axes so we will hide them. To hide
the Coordinate axes
axes, click the View menu on the menu bar, and then
click Axes.. To hide the Algebra window, click View then click Algebra
window.
2.) Click the Segment between two points tool and click two distinct
places on the drawing pad to construct segment AB.
3.) If the labels of the points are not displayed, click the Move button,
button
right click each point and click Show label from the context menu.
m
(The context menu is the pop-up
up menu that appears when you right click
an object.)
4.) To construct a circle with center A passing through B,, click
the Circle with Center through Point tool, click point A, then click
point B.. After step 4, your ddrawing
rawing should look like the one shown
in Figure 2.
Figure 2 - Circle with center A and passing through B
5.) To construct a line perpendicular to AB and passing through A, click
the Perpendicular line tool, click segment AB then click point A.
6.) To intersect the circle and the line, use the New Point tool and click
one of the intersections. If the label of the third point is not shown, right
click the point, then click Show label.. After step 6, your drawing should
look like the one shown in Figure 3.
Figure 3 - Circle with radius AB and a line perpendicular to AB passing
through A
7.) Next, we construct a line parallel to AB then passing through C. To
do this, click the inverted triangle at the bottom right of
the Perpendicular line tool, then choose the Parallel line tool. Click
segment AB,, then click point C.
8.) Now, we construct a line parallel to AC and passing through point B.
To do this, with the Parallel line tool still active, click line AC then
click point B.
9.) Intersect the line passing through point B and point C to form
point D,, the fourth vertex of the square. After step 9, your drawing
should look like the one shown in Figure 3.
Figure 3 - Square formed from radii a circle
10.) Using the Move button, move the point A or point B. What do you
observe?
11.) Next we hide all the objects except the four points and
segment AB.. To do this, right click each object and then uncheck Show
Object from the context menu.
12.) To complete the square, connect points A and C,, points C and D,
and points B and with the Segment between two points tool.
13. To verify that the quadrilateral that we created is indeed a square,
right click each side and click Object Properties.
14. In the Object Properties window, select the Basic tab. Be sure that
the Show label check box is checked and choose Value from the Show
label drop down list box. (see the last figure in Tutorial 2))
15. To measure angle A, click the Angle tool,, and click the vertices in
the following order: point C, point A and then point B.
Figure 4 - Square ABCD with a reflex angle on vertex A
16. In case the angle formed is a reflex angle, right click the symbol
(the green sector), click Object Properties from the context menu.
17. In the Basic tab of the Object Properties window, uncheck
the Allow Reflex Angle check box, and then click the Close button.
18. Reveal the measure of the three other angles. Move point A or
point B. What do you observe?
19. Explain why the construction above always results to a square.