Guess the Equation of the Line

Consider the graph of a linear function in the applet below. 

  1. Move sliders a and b to guess the equation of the graph. 
  2. Click the check box to check your answer. 
  3. Click the New Line button to generate a new line.

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Guillermo Bautista, Created with GeoGebra

Quadrilateral Angle Sum

Move points A, B, C or D to create the desired quadrilateral. Move the slider to the extreme right.  As each slider appears, rotate the quadrilateral by moving it to the extreme right. 
This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Guillermo Bautista, Created with GeoGebra

The applet above demonstrates that the angle sum of a quadrilateral is 360 degrees. 

Snapshot


Pythagorean Theorem Proof 2

Given: Triangle with sides a and b and hypotenuse c and three squares containing a, b and c.
 

Questions 
  1. What are the areas of the green square and the red square? 
  2. Move slider p to the extreme right. What types of quadrilaterals were formed from the two squares? 
  3. What can you say about the areas of squares in Question 1 and the areas  of quadrilaterals in Question 2?
  4. Move slider q to the extreme right. What type of quadrilaterals were formed? 
  5. What can you say about the quadrilaterals formed in Question 1 and the quadrilaterals in Question 4?
  6. Move slider r to the right. What do you observe?
  7. What can you say about the areas of the red square,  green square, and the area of square containing c?
  8. In terms of a, b, and c, what equation can be formed relating the areas of the three squares?
 

Pythagorean Theorem Proof 1

Move slider \alpha to the extreme right.
1.) What do you observe? What is the area of the larger square (green and red regions)?

 

2.) Move slider p to the extreme right. What is the area of the figure formed in terms of a and b?

3.) What can you say about the area of the square in Question 1 and the area of the figure in Question 2?

4.) What equation can you form using the areas of these squares?  

Deriving the Area of Parallelograms

Given below is a parallelogram with base b and height h. Move points A and B determine the shape and size of the parallelogram, and then move the red point to the extreme right and observe what happens.
 
Questions 

  1. If the red point is on the extreme right of the segment, what shape is formed? Justify your answer. 
  2. What is the area of the shape in Question 1? 
  3. How is the area of the shape in Question 1 related to the area of the original figure?
  4. What is the area of the parallelogram in terms of b and h
  5. What can you conclude about the area of the parallelogram in the original figure and the area of the shape in Question1? 
 

Deriving the Area of Trapezoid

Given below is a trapezoid with bases b1 and b2 and height h. Move the slider to the extreme right and observe what happens. 


Questions

  1. When alpha is 180 degrees, what shape is formed? Justify your answer. 
  2. How is the area of the shape in 1 computed based on the figure? 
  3. What is the relationship between the area of the shape in 1 and the area of the trapezoid?
  4. What formula will describe the area of the trapezoid based on the given above? 

A Visual Proof of the Pythagorean Theorem

Move point E to determine the shape of the triangles, and then move slider α to the extreme right.
This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Guillermo Bautista , September 11, 2012, Created with GeoGebra

Questions:

  1. What is the relationship among the areas of squares with side lengths a, b, and c
  2. What equation describes the relationship among the areas of the three squares? 
  3. Prove that the green quadrilateral inside the large when  α is  90° is a square
Snapshot

Reflecting the Exponential Function

The red point is the reflection of the green point on the blue line. Move the green point and observe what happens.

Questions

  1. What is the relationship between the coordinates of the points?
  2. If the coordinates of the green point are (x,y), what will be the coordinates of the reflected point?
  3. If the green point and red point are connected with a line, what can you say about that line and the line y = x?
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