The Elder Wand

1. 

   Steven Swift

   September 12, 2011 at 2:01 pm

   Let me be the first to say the obvious (to Math Nerds), its the Altitude, aka the Height.

   Now, keeping it straight up, slide that dotted line left or right along the base, and reconnect the top tip to wherever it is. How does that change the area of the triangle? And can u prove it?
2. 

   Mike

   September 12, 2011 at 2:12 pm

   @ The above comment: It’s not JUST an altitude. It’s also a: perpendicular bisector, angle bisector, and a median. In fact, given the fact that the right angle was made explicit, I’m guessing the preferred answer was perpendicular bisector.

   As for your question, the are is unaffected. Given the area of a triangle being 1/2 * (b*h) with b being the length of the side with which the altitude intersects (Or, possibly, would intersect if the side was extended), and h being the length of the altitude. Neither are varying at all, so the area must stay the same
3. 

   Steven Swift

   September 13, 2011 at 9:47 am

   Correct Mike! Although, we don’t “know” that the round-looking thing is a perfect circle and we don’t know for sure if the altitude line is exactly in the center of the base. It is clearly labelled perpendicular, but might not be a bisector or angle bisector. But that is nit-picky…

   A proof that the area is unaffected by sliding the altitude left/right is very simple, but an excellent demonstration for young people and geometric proof newbies, that is why I challenged it. Hint: draw two rectangles, one on the left and one on the right of the altitude, same height and to the left or right edge of the base. Compare the area of the two triangles inside the two rectangles… Start your kids thinking along these lines, its a good mental exercise to start off geometric reasoning!

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