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        riddles >> medium >> A circle with center O
        
(Message started by: tony123 on Oct 4^th, 2007, 2:58am)

Title: A circle with center O
Post by tony123 on Oct 4^th, 2007, 2:58am

A circle with center O and radius 1 cm rolls around the inside of a triangle whose sides are 6, 8, and 10 cm, always touching one or more of the sides as it rolls. How far does O travel in one complete circuit?

Title: Re: A circle with center O
Post by towr on Oct 4^th, 2007, 5:56am

Seems easy to me, you can just shave the edges off, then walk around the perimeter as a point.

Title: Re: A circle with center O
Post by FiBsTeR on Oct 4^th, 2007, 6:25pm

If the original triangle has vertices {(0,0),(0,6),(8,0)}, then O will travel [hide]along the perimeter of the triangle with vertices {(1,1),(1,[21/4]),([20/3],1)}; this perimeter is equal to 17.[/hide]

EXTENSION: Let A be the original triangle above, with vertices at {(0,0),(0,6),(8,0)}, and B be the triangle with vertices at [hide]{(1,1),(1,[21/4]),([20/3],1)}[/hide]. Suppose point P starts at (0,0) on A and point Q starts at [hide](1,1)[/hide] on B, both moving 1/12 unit per second clockwise around the perimeter of their respective triangles. How much time will elapse until both points are once again on a vertex of their triangles at the same time?

Title: Re: A circle with center O
Post by FiBsTeR on Oct 8^th, 2007, 11:18am

Well, since this got buried unanswered, the answer to my question was [hide]1224 seconds or 20.4 minutes[/hide]. :P