If (-5,3) and (5,3) are two vertices of an equilateral triangle, then find co- ordinates of the third vertex, given that origin lies inside the…

\[\begin{aligned} & A(-5,3) \quad B(5,3) \quad C(x, y) \\ & A B=10=A C \\ & A C^{2}=100 \\ & (-5-x)^{2}+(3-y)^{2}=(5-x)^{2}+(3-y)^{2} \\ & 20 x=0 \\ & x=0 \\ & (3-y)^{2}=75 \\ & 3-y= \pm 5 \sqrt{ } 3 \\ & y=3-5 \sqrt{ } 3 \\ & y=-5.5 \end{aligned} \] The coordinates of the third vertex are $\displaystyle (0,-5.5)$