# The base of an equilateral triangle is x + y - 2 = 0 and the opposite vertex Is (2,-1) find the equations of the remaining sides

Vikas TU
14149 Points
8 years ago
I am writing the other vertices of the triangle which are lying on the given eqn.=0 in parametric form of eqn.x + y - 2
as:
(x1, 20- x1)  and (x2, 20 – x2).

let a =  distance b/w (2, -1) and (x1, 20- x1) points,
b =  distance b/w (2, -1) and (x2, 20- x2) points,
c =  distance b/w (x2, 20-x2) and (x1, 20- x1) points,

Apply distance formulae , and equate the thre distances and solve three eqns. as it is an equilateral triangle.
Therfore, doing so
U will get two of the eqns as:

2x1^2 – 4x1x2 – 46x2 – 445 = 0
2x2^2 – 4x1x2 + 46x1 – 445 = 0
solving both of them,

we get,

(x1 + x2) (x1 – x2) – 23(x1 + x2) = 0
(x1 + x2) (x1 – x2  - 23) = 0.........................................A
x1 + x2 = 0  and x1 – x2 =  23 …..........................B
solving eqns. A and B -
x1 = 11.5  and x2 = -11.5
and therfore the two other vertics are (11.5, 8.5)  and (-11.5 and 31.5) and the given vertex is (2,-1).
Hence now U can find the eqns. from these verticses.

Hope it Helps!