Regular Polygon, Apothem, Distance
In a regular n-sided polygon, the sum of the perpendicular distances from an interior point to the n sides being n times the apothem of the polygon.
See complete Problem 225 at:
gogeometry.com/problem/p225_viviani_theorem_regular_polygon.htm
Level: High School, SAT Prep, College geometry
Wednesday, January 14, 2009
Elearn Geometry Problem 225: Viviani Theorem Extension
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One can prove that, in a equiangled polygon, the sum of the distances of an arbitrary inner point to the sides is constant =]
ReplyDelete(1)let the no. of sides be n.
ReplyDelete(2)let the length of each side be y.
(3)therefore the area of the given polygon is 1/2ap.y.n
(4)from point P, we can do tringulations (on the sides)
(5)We note that sum of area of those triangls=area by 1/2 ap.y.n method =(h1+h2+h3+....+hn).y.1/2
(6)then, 1/2& y get cancelled & we get ap.n = h1+h2+h3+...+hn
(7)hence the proof!!