Question

if three points (0,0) (3,root 3) and (3,k) form an equilateral triangle then k=?

Answer 1

Step-by-step explanation:

vertex A (0,0)

vertex B (3,root 3)

vertex C ( 3,k)

side of equilateral triangle be same

$ab {}^{2} = {bc}^{2}$

(3-0)^2+ (√3-0)^2 = (3-3)^2+ (√3-k)^2

9+3= 0+3+k^2-2√3k

solve equation

Answer 2

Answer:  The required value of k is √3 or -√3.

Step-by-step explanation: Given that the three points  (0, 0) (3, √3) and (3, k) form an equilateral triangle.

We are to find the value of k.

We know that

the lengths of all the three sides of an equilateral triangle are equal.

So, according to the given information, we have

$\textup{distance between (0, 0) and }(3,\sqrt3)=\textup{distance between (0, 0) and (3, k)}\\\\\Rightarrow \sqrt{(3-0)^2+(\sqrt3-0)^2}=\sqrt{(3-0)^2+(k-0)^2}\\\\\Rightarrow (3-0)^2+(\sqrt3-0)^2=(3-0)^2+(k-0)^2~~~~~~~~~~~~~~[\textup{Squaring both sides}]\\\\\Rightarrow 9+3=9+k^2\\\\\Rightarrow k^2=3\\\\\Rightarrow k=\pm\sqrt3.$

Thus, the required value of k is √3 or -√3.