Problem:
A line that passes through the origin intersects both the line x=1 and the line y=1+33​​x. 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+33​​x. 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​).
