Question

If the points $(2,4),(2,6)$ and $(2+\sqrt{3},k)$ are the vertices of an equilateral triangle, then what is the value of $k$?

• A. $6$
• B. $5$
• C. $-3$
• D. $1$

Answer 1

Answer: (B)

$5$

Let the points be $A(2,4),B(2,6)$ and $C(2+\sqrt{3},k)$
All the sides of an equilateral triangles are equal.
So, $AB=AC$
$\Rightarrow {(2-2)}^2+{(4-6)}^2={(2-(2+\sqrt{3}\left)\right)}^2+{(4-k)}^2$

$\Rightarrow 4=3+{(4-k)}^2$

$\Rightarrow 1={(4-k)}^2$

$\Rightarrow 1=4-k$ or $-1=4-k$

$\Rightarrow k=3$ or $k=5$
Hence, option B is correct