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in Mathematics by (53.1k points)

ABC is an equilateral triangle in which B = (1, 3) and C = (−2, 7). Find the coordinates of the vertex A.

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See Fig. The slope of BC is

7 - 3/-2 - 1 = -4/3

Therefore, the slope of perpendicular bisector of BC is 3/4 and its equation is 

y - 5 = 3/4(x + 1/2)

where M(−1/2, 5) is the midpoint of BC. Since

BC = √(7 - 3)2 + (-2 - 1)2 = 5, we have

Altitude AM = (sin 60°)AB = 5√3/2

The equation of the altitude AM is

y = 3/8(2x + 1) + 5   ...(1)

Now,

75/4 = (AM)2

when x  = 2√3 - (1/2) we have

and when x  = -2√3 - (1/2) we have

y = 5 - 3√3/2

Hence

Aliter (Using Complex Numbers): B = 1 + 3i and C = −2 + 7i. Rotate CB about C through an angle 60° in anticlockwise sense (see Fig.). Thus

Similarly, if we rotate CB in clockwise sense about C through an angle 60°, we get the second position of A.

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