The Area of the Median Triangle - Dynamic Proof

Let ABC be a given triangle. The triangle with sides equal to the medians of ABC is called the Median Triangle.
We will show that the area of the median triangle is 3/4 of the area of the original triangle ABC.

•  Drag the two sliders to the end to construct the median triangle by translating the medians.

To rearrange the area of the median triangle:

• Click the Rearrange checkbox. This will show a green point at the vertex of the median triangle. 
•  Drag the green point to C.
•  Drag the green point at B to M_c. 
• Drag the green point at M_a to M_c. 

Irina Boyadzhiev, 29 January 2014, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Pythagorean Theorem

Irina Boyadzhiev, 13 December 2013, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Vertical Line Test

This applet can be used to determine whether one relation is a function or not by using the Vertical Line Test. The Vertical Line Test says that if some vertilcal line meets the graph in more than one point then the relation is not a function.

•  In the input box enter a polynomial equation of x and y in the form "polynomial = number".
  For example, to graph y = x², enter y - x² = 0 
• Drag the slider to use the vertical line test.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Inverse Function

This applet can be used to study inverse functions.

• Enter a function of x in the input box f(x).
• Set the domain of the function by dragging the endpoints of the blue line at the bottom of the window.
• Click on the “Horizontal Line Test” and drag the vertical slider. Is f(x) a one-to-one function?
• Click on the “Reflect f(x)” button.
• Drag the horizontal slider to run the Vertical Line Test. Is the reflection a graph of a function?
• Drag the endpoints of the blue segment under the graphs to restrict the domain of f(x) in such a way that the reflected graph is a graph of a function.
• Hide all points created by the horizontal and vertical line tests by clicking the respective checkboxes.
• Click the “Show Corresponding Points”. Drag the orange point along the graph of f(x) and notice its reflection on the graph of the inverse function.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

Download applet 

Irina Boyadzhiev's  GeoGebra Applets