Q. 153.8( 23 Votes )

If two vertices of an equilateral triangle are (0, 0) and then find the third vertex.

Answer :

Given two vertices of an equilateral traingle ABC


A (0, 0)


B (3, )


Let the third vertex be C (x, y)


We know that distance of a point A (x,y) from origin O (0, 0) is given as OA =


So,






Using distance formula,





Since ∆ABC is an equilateral, AB = BC = CA.


Using CA = AB,



Squaring both sides, we get


x2 + y2 = 12 …(i)


Now, using AB = BC



(Using eq (i))


Squaring both sides, we get


– 6x – 2√3y + 24 = 12


6x + 2√3y = 12


3x + √3y = 6



Substituting the value of x in eq (i), we get





12y2 – 12√3y +36 = 108


12y2 – 12√3y – 72 = 0
y2 – √3y – 6 = 0


y2 – 2√3y + √3y – 6 = 0


y (y – 2√3) + √3 (y – 2√3) = 0


(y – 2√3) (y + √3) = 0
y = 2√3 or – √3
If y = 2√3, then

If y = –√3,

So, the third vertex of the equilateral triangle = (0, 2√3) or (3, –√3).


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