## The Dancing Triangle and Its Applications

In the figure below, lines *l* and *m* are parallel lines. What can you say about the areas of triangle *ABC* and triangle *ADC*?

The distance between two parallel lines is equal at any point, so the two triangles have the same altitude (can you see why?). Further, the two triangles have a common base, therefore, their base lengths are equal. So, the areas of the two triangles are equal. In fact, you can choose any point *P* on line *l* and the areas of the triangle *ACP* will always equal to the areas of triangles *ABC* and *ADC*. We like to call this triangle the dancing triangle because using an applet, you can dance it by moving *P* without changing the area. In the applet below, move points *B* and *D* to dance the triangle. » Read more