Problem:
In the figure shown below, ABCDE is a regular pentagon and AG=1. What is FG+JH+CD?
Answer Choices:
A. 3
B. 12−45​
C. 35+25​​
D. 1+5​
E. 1011+115​​
Solution:
Triangles AGB and CHJ are isosceles and congruent, so AG=HC=HJ=1. Triangles AFG and BGH are congruent, so FG=GH. Triangles AGF,AHJ, and ACD are similar, so ba​=ca+b​=d2a+b​.
Because a=c=1, the first equation becomes b1​=11+b​ or b2+b−1=0, so b=2−1+5​​. Substituting this in the second equation gives d=21+5​​, so b+c+d=(D)1+5​​.
