Welcome to the spoken tutorial on Theorems in GeoGebra.
In this tutorial we will state and prove, Pythagoras theorem and Midpoint theorem using Geogebra .
To record this tutorial, I am using; Ubuntu Linux OS version 16.04 GeoGebra version 5.0.438.0-d.
To follow this tutorial, learner should be familiar with GeoGebra interface. For the prerequisite GeoGebra tutorials, please visit this website.
Let us state the Pythagoras theorem.
The square of the hypotenuse (the side opposite to the right angle) is equal to the sum of the squares of the other two sides.
I have already opened the GeoGebra interface.
We will begin with the drawing of a semicircle.
Click on the Semicircle through 2 Points tool.
Then click to mark two points in the Graphics view.
Using the Point we will mark another point C on the semicircle c.
Now let us draw a triangle ABC using the points on the semicircle.
Click on the Polygon tool and draw triangle ABC
Here we are using semicircle to draw the triangle.
This is because we need the measure of one angle to be 90 degree.
Let us measure the angles of the triangle.
Click on the Angle tool and click inside the triangle. Here angle ACB is 90 degrees.
Now we will hide the semicircle c.
In the Algebra view under Conic, click on the blue dot against c.
We will draw three squares using the sides of the triangle.
For that click on the Regular Polygon tool and then click on the points C, B.
The Regular Polygon text box opens with a default value 4.
Click on OK button at the bottom.
If you click on the points B, C, the square is drawn in the opposite direction.
Let us undo the process by clicking on the Undo button.
Now click on the points A, C. And then click the OK button in the text box that appears.
Similarly click on the points B, A. And then click the OK button in the text box that appears.
Now we have three squares that represent the Pythagorean triplets.
Now we will use Zoom Out tool to see the diagram clearly.
Now we will find the area of these squares.
Click on the Area tool and click on poly1, poly2 and poly3 respectively.
The areas of the respective squares are displayed.
Using the Move tool drag the labels to see them clearly.
Now we will check if the area of poly1 + area of poly 2 is equal to area of poly3.
In the input bar type poly1+ poly2 and press Enter.
In the Algebra view a Number d, shows the value of area of poly3.
Hence Pythagoras theorem has been proved.
Now I will explain the Construction Protocol for pythagoras theorem.
Construction Protocol shows the step by step construction of the diagram as an animation.
To view the animation, click on View menu and select Construction Protocol check box.
Construction Protocol view opens next to Graphics view.
I will drag the boundary of Graphics view view to see the Construction Protocol view.
This view has a table with some columns. Below the table we have the animation controls.
Now click on the Play button.
Watch the step by step construction of the figure as an animation.
Now we will prove the Mid-point theorem.
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
I have opened a new GeoGebra window.
Let us draw a triangle ABC using Polygon tool.
Now we will find the mid-points of the sides AB and AC.
Click on the Midpoint or Center tool. Then click on the sides AB and AC.
Using the Line tool, draw a line through points D and E.
Now we will draw a line parallel to segment AB.
For this, click on the Parallel Line tool and click on segment AB.
Then, click on point C. Line g parallel to segment AB is drawn.
Notice that lines f and g intersect at a point.
Using the Intersect tool, let us mark the point of intersection as F.
Now we need to measure angles F C E and D A E.
Click on the Angle tool and click on the points F, C, E and D, A, E.
Notice that angles are equal since they are alternate interior angles.
Similarly we will measure C, B, D and E, D, A .
The angles are equal. This implies that line f is parallel to segment BC.
Using the Distance or Length tool, click on the points D, E and B, C. Notice that DE is half of BC.
Hence the mid-point theorem is proved.
Once again I will show the Construction Protocol for the theorem.
Click on View menu select Construction Protocol check box.
Construction Protocol' view opens next to Graphics view.
Now click on the Play button. Watch the step by step construction of the figure.
As an assignment, prove this theorem.
Your completed assignment should look like this.
Let us summarize what we have learnt.
In this tutorial we stated and proved, Pythagoras theorem and Midpoint theorem using Geogebra
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This is Madhuri Ganapathi from, IIT Bombay signing off. Thank you for watching.