Thursday, July 29, 2010

Triangle Area - Quasi-Simple

Continuing with the previous triangle area thread, here is a problem that is slightly more complex / interesting: How do you model the changing area of the outer (blue) triangle over time, as point B rotates around the circle?

Here is the screenshot, but again I would advise you to check out the actual GeoGebra animation (linked to above) to get a visualization of what we're dealing with.

Fortunately, as you can well imagine, the two areas are intimately related. By examination, the two triangles (red and blue) share, in fact, a common altitude and their bases are related by a factor of 1.5. Thus, we can piggy-back on top of results for the simpler case (see entry from two days back) and, in essence, the two graphs for the two triangles will look like mere scaled versions of one another:

Beautifully simple, no? (As this can be extended to show that any such blue/outer triangle will have an area that is a scaled version of the red/inner triangle.) Now, off to thinking about how this changes if two of the three vertices lie on the same circle, but rotate at different rates...

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By the way, here is a cutesy SAT problem, good for your middle-schoolers or anyone learning about rise / run (courtesy of David Marain):
Points A(4,5), B(7,9) and C(t,u) are on a line so that B is between A and C and BC = 5(AB). What is the value of u?
For high-schoolers, you can extend this problem a bit to ask where C would be located if BC = n(AB).