The interactive models below accompany the article "Theorems in Motion: Using Dynamic Geometry to Gain Fresh Insights" by Daniel Scher in the April 1996 issue of the Mathematics Teacher. The article shows how dynamic geometry software can be used to construct constant perimeter and constant area rectangles. We define these terms as follows:
In the interactive model below, point C divides length AB into segments AC and CB. Drag point C back and forth along AB or press the 'Animate' button. Notice that the length and width of the accompanying rectangle adjust themselves to remain equal to AC and CB. Since AC + CB is a constant, the perimeter of the rectangle is also constant. To change the value of the perimeter, drag point B on the red/blue segment.
The interactive model below shows a circle along with chord AB composed of segments AP and PB. Drag point A around the circumference of the circle (or press the 'Spin!' button). Notice that chord AB pivots around the fixed point P within the circle. The Power-of-a-Point theorem tells us that the product AP x PB remains constant as the chord pivots.
The length and width of the accompanying rectangle are constructed to be equal to AP and PB. As the chord spins, the dimensions of the rectangle adjust themselves to remain equal to these segments. Thus the area of the rectangle is the constant product AP x PB.
Try moving point P to different locations to change the constant area of the rectangle.
Using this method, is it possible to construct all rectangles with a given constant area?
What kind of curve is being traced by the upper right vertex of the rectangle?
The interactive model below shows two parallel lines and right triangle ABC constructed between them. Segment BD is an altitude of the triangle. Drag point B to the right and left along its segments. You'll generate a whole collection of right triangles, all of which have the same altitude length, BD.
When moving point B, notice that the dimensions of the accompanying rectangle adjust themselves to remain equal to AD and DC. The geometric mean theorem tells us that the product AD x DC is equal to the square of BD. Since BD maintains a constant length, AD x DC is also constant. Thus the area of the rectangle is constant.
Does this method generate a larger collection of rectangles than the spinning chord construction above?