Problem: If ΞA1βA2βA3β is equilateral and An+3β is the midpoint of line segment AnβAn+1β for all positive integers n, then the measure of X44βA45βA43β equals
Answer Choices:
A. 30β
B. 45β
C. 60β
D. 90β
E. 120β
Solution:
Triangle A2βA3βA4β has vertex angles 60β,30β. 90β, respectively. Since β A1βA2βA3β=60β, and A2βA4β and A2βA5β have the same length, ΞA2βA4βA5β is equilateral. Therefore, ΞA3βA4βA5β has vertex angles 30β,30β,120β. respectively. Then ΞA4βA5βA6β has vertex angles 30β,60β,90β, respectively.
Finally, since A4βA5βA6β=60β and A5βA6β and A5βA7β have the same length, β³A5βA6βA7β is again equilateral. Therefore β³AnβAn+1βAn+2β βΌΞAn+4βAn+5βAn+6β with Anβ and An+4β as corresponding vertices. Thus β A44βA45βA43β=β A4βA5βA3β=120β.
