The Midline Theorem

Two midpoints of a triangle are connected with a segment. Investigate.

1.) What did you observe in your investigation?

2.) What do your observations about the measures of the angles mean? How do they relate to the properties of the segments? 

3.) What did you observe about the lengths of segment DE and segment BC? Was this observation always true?

4.) Based on the discussion, make conjectures about the relationship between segment AB and segment BC.

The Pantograph

A pantograph is a mechanical linkage connected in a special manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged or miniaturized copy will be drawn by a pen fixed to the other. The pantograph was invented by Cristoph Scheiner.

The applet above demonstrates how the pantograph works.  Move the blue point (hinge) and draw any shape. 

1.) What do you observe?
2.) What is the relationship between the blue figure and red figure?
3.) Justify your answer in 2.
4.) Prove that the relationship in 2 always holds.

Approximating Pi

Move slider n to the right and observe what happens to the inscribed polygon.
The applet above shows the relationship between the area of a circle and the inscribed polygon.  Notice that as the the number of sides (n) of the inscribed polygon increases, its area gets closer and closer to the area of the circle which is equal to .  This approximation is called the method of exhaustion. It was developed used by Archimedes.