Pythagorean Theorem Exploration
In this activity, we will explore the meaning of the formula from the Pythagorean Theorem through a geometrical construction like the one below.
Let's get started!
First, choose the text tool and click on the construction area to open a text box. Title your sketch The Pythagorean Theorem: .
In order to see what the title will look like—which is helpful when writing formulas—use the extended text box by clicking on the Advanced option.
Let's get started!
First, choose the text tool and click on the construction area to open a text box. Title your sketch The Pythagorean Theorem: .
In order to see what the title will look like—which is helpful when writing formulas—use the extended text box by clicking on the Advanced option.Creating a Square
1. On the toolbar, select the segment tool. You may construct one on the work area by clicking on two separate places that will become the ends of the segment. You may then use the arrow (move) tool to click on an endpoint and make the segment larger or smaller.
2. Label the endpoints T and R. Do this by clicking on the text tool and clicking on the endpoints of the segment. Most likely, the labels were A and B. You can relabel them T and R by right-clicking on the label and accessing the settings and changing the label name at the top of the menu. This box allows you to type in any label you wish–including phrases! Try a few and also try out the settings menu to try different colors and styles. Changes are saved automatically when you exit this menu.
3. There are many ways to construct a square. We will construct it using rotation. Select the
Rotate around Point tool from the transformations section of the menu. Click on point T to select it as the rotating point and then on R to select it as the center of rotation. Type in 90 degrees. Repeat using a new center of rotation until a square results.
4. Construct the interior of the square by clicking on the vertices with the Polygon tool selected. Note that you will need to click on the first vertex again to close the shape for GeoGebra to understand that the polygon is finished.
Note: Alternatively, you can select the Regular Polygon tool and create a segment on the work area. Then, type the number of vertices you want your polygon to have (4 for a square) and the shape will be constructed automatically.
Constructing a Right Triangle
1. Construct a segment. Select the Perpendicular Line tool and click on an endpoint and the segment.
2. Place a point on this line by selecting the point tool and clicking where you want to place it. Connect it with the other endpoint of the segment.
3. Construct all the segments that make the sides of the triangle by selecting the segment tool and clicking on each pair of vertices.
4. Select the Show/Hide Object tool and click on the perpendicular line to hide it.
5. Use the Polygon tool and click on the vertices of the triangle to construct its interior.
6. Select the Distance or Length tool from the Measure section of the menu and click on each side of the triangle for their lengths to appear.
7. Construct a square on a side of the triangle. You should get a square whose side lengths equal the length of the side between the vertices selected. Repeat this for all 3 sides of the triangle.
Comparing Areas
1. Measure the area of each square by using the Area tool in the Measure section of the
menu.
2. Use the Algebra menu to calculate the sum of the areas of the two small squares. (In
the example, we called the squares sq2 and sq3 so we just had to type sq2 + sq3). Then,
click and drag the result to the work area for it to appear next to the construction.
3. Under the Algebra menu again, use the calculator to square the length of each side.
4. Re-label each as “a squared,” “b squared,” and “c squared” as appropriate. Do this by
right-clicking on the three dots by the calculation to open a menu and select Settings.
5. Move the triangle around and change its size by using the Move tool on one of the vertices of the original triangle. Notice how the Pythagorean Theorem always holds!!
6. Choose the Text tool and click anywhere in the sketch to insert a text box. Answer the following question: What does the formula mean? Use your picture to help you explain. (Focus on describing the areas of the squares along the sides of your triangle and how they are related.)
3. Under the Algebra menu again, use the calculator to square the length of each side.
4. Re-label each as “a squared,” “b squared,” and “c squared” as appropriate. Do this by
right-clicking on the three dots by the calculation to open a menu and select Settings.
5. Move the triangle around and change its size by using the Move tool on one of the vertices of the original triangle. Notice how the Pythagorean Theorem always holds!!
6. Choose the Text tool and click anywhere in the sketch to insert a text box. Answer the following question: What does the formula mean? Use your picture to help you explain. (Focus on describing the areas of the squares along the sides of your triangle and how they are related.)