co-ordinate

Prove that (2a,4a), (2a,6a) and (2a+a root3,5a) are vertices of an equilateral triangle.

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  1. just show that the sides have the same length.

    For instance, the first two points define a side of length d, where

    d^2 = (2a-2a)^2+(6a-4a)^2 = 4a^2
    so, d=2a

    Now just show that the other two sides are the same length.

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  2. We could "shrink" the figure, after all sides of all equilateral triangles are in the same ratio, so let's use
    A(2,4), B(2,6), and C(2+√3 , 5)

    are the angles equal to 60° each ??

    clearly AB is a vertical line.
    slope CA = 1/(2+√3 - 2) = 1/√3
    so CA makes an angle of 60° with the x-axis
    slope CB = -1/√3, so CB makes an angle of 120°

    So , look at your sketch, what do your think ?

    Btw, we could have found the slopes using the original points containing the "a" 's, they would have cancelled .

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