Question

If $\left(-4,3\right)\mathrm{and}(4,3)$ are two vertices of an equilateral triangle , find the coordinates of the third vertex , given that the origin lies in the
(i) interior (ii) exterior of the triangle .

Answer 1

Suppose A(−4, 3) and B(4, 3) be the vertices of an equilateral triangle.
The distance AB is
$AB=\sqrt{\left(4-\left(-4\right)\right)+{\left(3-3\right)}^2}=8units$
Consider the figure below:


Here, D is point such that CD is perpendicular to AB.
Thus, AD = 4 units.
By pythagoras theorem, we have
$$\begin{gathered} A{D}^2+C{D}^2=A{C}^2 \\ \Rightarrow 4^2+C{D}^2=8^2 \\ \Rightarrow C{D}^2=64-16 \\ \Rightarrow C{D}^2=48 \\ \Rightarrow CD=4\sqrt{3} \end{gathered}$$

Since C is on the negative y axis, so coordinate of C are $3-4\sqrt{3}$ units.
If C is on the positive y axis, so coordinate of C are $3+4\sqrt{3}$ units.