Problem:
A line that passes through the origin intersects both the line x=1 and the line y=1+33x. The three lines create an equilateral triangle. What is the perimeter of the triangle?
Answer Choices:
A. 26
B. 2+23
C. 6
D. 3+23
E. 6+33
Solution:
Label the vertices of the equilateral triangle A,B, and C so that A is on the line x=1 and B is on both lines x=1 and y=1+33x. Then B=(1,1+33). Let O be the origin and D=(1,0). Because △ABC is equilateral, ∠CAB=60∘, and △OAD is a 30−60−90∘ triangle. Because OD=1,AD=33 and AB=AD+DB=33+(1+33)=1+323. The perimeter of △ABC is 3⋅AB=3+23. Indeed, △ABC is equilateral with C=(−23,21).
