GeoGebra Tutorial 5 – Discovering the Pythagorean Theorem
In this tutorial, we are going to compare areas of squares formed from sides of a right
triangle. To construct this figure, we first construct a right triangle, and form three
squares, each of which contains one of the three sides as shown below. Then we will
observe the relationship among the areas of these squares.
Figure 1 - Squares formed containing the sides of a right triangle
In this tutorial, we are going to learn how to use the Regular Polygon tool.
constructions that you will make here are review of the first four tutorials.
START
Most of the
1.) Open the GeoGebra window. 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 menu that appears when
you right click an object.)
4.) Next, we will construct a line perpendicular to segment AB and passing through point B. To do
this, choose the Perpendicular line tool, click segment AB, then click point B.
5.) Next, we create point C on the line. To do this, click the New point tool and click somewhere on
the line. Your drawing should look like the figure below. Display the label of the point in case it is
not shown (see no. 3)
Figure 2 - Point C on the line passing through B
You have to be sure that C is on the line passing through B. That is, be sure that you cannot drag
point C out of the line. Otherwise, you have to delete the point and create a new point C.
6.) Hide everything except the three points. To hide the line, right click the line and uncheck Show
Object. Do this, also, to segment AB.
7.) Next, we rename point B to point C and vice versa. To rename point B toC, right click point B,
click Rename and then type the new name, in this case point C, in the Rename text box, then click
the OK button. Now, rename B (or B1) to C.
8.) Next we construct a square with side AC. Click the Regular polygon tool, then click
point C and click point A.
9.) In the Points text box of the Regular polygon tool, type 4 since we are going to create a square.
If the position of the square is displayed the wrong way (right hand side of AC) just click
the Edit menu, click the Undo button and reverse the order of the clicks.
Figure 3 - Square containing side AC
10. ) With the Polygon tool still active, click point B and click point C to create a square with
side BC. Similarly, click point A, then click point B to create a square with side AB. After step 10,
your drawing should look like the one shown below.
Figure 4 - Squares containing sides of right triangle ABC
11.) Hide the label of the sides of the squares by right-clicking them, then un-checking Show label.
12.) Rename the sides of the rectangle as shown below.
Figure 5 - Triangle ABC with side lengths a, b and c.
13.) Now, let us reveal the area of the three squares. Right click the interior of square with side AC,
then click Object Properties from the context menu.
14.) In the Basic tab of the Object Properties window, check the Show Label check box and
choose Name & Value from the drop-down list box. Do this to other squares as well. You can
click poly1, poly2 and poly3 from the Object list. These are the squares.
Figure 6 - Properties of squares shown in the Object Properties window
15.) Move the vertices of the triangle. What do you observe about the area of the squares?
16.) You may have observed that the area of the biggest square is equal to the sum of the areas of
the two smaller squares. To verify this, we can put a label in the GeoGebra window displaying the
areas of the three squares.
17.) Suppose the side of the two smaller squares are a and b, and the side of the biggest square is c,
what equation can you make to express the relationship of the of the three squares?
18.) What conjecture can you make based on your observation?
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