A line that passes through the origin intersects both the line x = 1

Question

A line that passes through the origin intersects both the line x = 1 and the line  y=1+ \frac{\sqrt{3}}{3} x . The three lines create an equilateral triangle. What is the perimeter of the triangle?

(A)  2\sqrt{6}

(B)  2 + 2\sqrt{3}

(C) 6

(D)  3 + 2\sqrt{3}

(E)  6 + \frac{\sqrt{3}}{3}

bidskamikaze · Accepted Answer

Let the line that passes thru origin be line L. This line intersects two lines: X=1 and Y= 1+\sqrt{3}/3 x Note that the line Y= 1+\sqrt{3}/3 x makes 30 deg angle with horizontal since the slope is 1/\sqrt{3} This makes it impossible for L to intersect the two lines in the first quadrant

The three lines will intersect in a manner shown in the image. We are required to find the perimeter of equilateral triangle ABC. The line Y= 1+\sqrt{3}/3 x intersects Y axis at 1 (point D) (The origin of coordinate system is at O) Now consider right triangle POB: Angle PBO=60 and angle BPO=90. So, angle BOP=30. This makes it a 30-60-90 triangle where ratio PB:PO:BO will be 1:\sqrt{3}:2 . We know that PO=1. Therefore PB= 1/\sqrt{3} , BO= 2/\sqrt{3} . Now consider triangle COD: Angle OCD=60. Also, angle DOC and CDO=60 and hence triangle COD is also an equilateral triangle. We know that DO=1. Since COD is an equilateral triangle, DO=CD=CO=1. We now already know the length of side CB of equilateral triangle ABC

CB=CO+BO=1+ 2/\sqrt{3} As the triangle is equilateral, AB=BC=AC. Perimeter= 3*(1+ 2/\sqrt{3}) = 3+2 \sqrt{3} Answer: D (Kudos if you liked the post

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pandajee · Answer

Please find my way of solving the problem. It's based on rough drawing of all the points and lines.

nick1816 · Answer

two lines are making 60 degrees angle with line x=1 and all three lines create an equilateral triangle, that means if one line has +ve slope, other line must have -ve slope. 
Equation of a line, having -ve slope, passing through origin and making 60 degrees with line x=1 is
y=-(1/√3)x
Intersection point of x=1 and y=-(1/√3)x= (1, -1/√3)
Intersection point of x=1 and y=1+(√3/3)x= {1, 1+(√3/3)}
distance between (1, -1/√3) and {1, 1+(√3/3)}= (3+2√3)/3
perimeter of equilateral triangle= 3*side = 3* (3+2√3)/3 = (3+2√3)

Crytiocanalyst · Answer

2 lines make 60 degree angle one has positive slope and other negative slope 
m=-1/3^(1/2) since 60 degree and negative slope 
c=0 passing through origin 
y=-(1/√3)x
The meeting of the lines of:
x=1 and y=-(1/√3)x= (1, -1/√3)
The meeting of the lines of:
x=1 and y=1+(√3/3)x= (1, 1+(√3/3))
one side of the equilateral triangle being the distance between 
(1, -1/√3) and (1, 1+(√3/3))= (3+2√3)/3
perimeter of the triangle being=3* (3+2√3)/3 = (3+2√3)(equilateral triangle )
Hence IMO D

bumpbot · Answer

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A line that passes through the origin intersects both the line x = 1

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A line that passes through the origin intersects both the line x = 1

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