A pentagon is such that each triangle formed by three adjacent vertices has area 1. Find its area.
ratio of short side to longer side of the triangle so formed:
x*sqrt(1/5(5+2sqrt(5)))x=1 area of triangle=1/2 base*height*2 such triangles
x=(5-2sqrt(5))^(1/4) length of short side
sqrt(x^2+(sqrt(1/5(5+2sqrt(5)))x)^2 = sqrt(2+2/sqrt(5))x length of hypotenuse (=side of the pentagon)
sqrt(2+2/sqrt(5))(5-2sqrt(5))^(1/4) = 2^(3/4)(1-1/sqrt(5))^(1/4) substituting for x
area of pentagon of side 2^(3/4)(1-1/sqrt(5))^(1/4)=sqrt(5/2(3+sqrt(5))
agreeing numerically with Larry's solution
Edited on February 14, 2025, 10:24 pm
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Posted by broll
on 2025-02-14 22:23:54 |
doing it the hard way
