First slide

Cartesian plane

Question

The incentre of the triangle with vertices $(1, \sqrt{3}), (0, 0)$ and $(2, 0)$ is

Moderate

A

$(1, \frac{\sqrt{3}}{2})$
B

$(\frac{2}{3}, \frac{1}{\sqrt{3}})$
C

$(\frac{2}{3}, \frac{\sqrt{3}}{2})$
D

$(1, \frac{1}{\sqrt{3}})$

Solution

Let $A (1, \sqrt{3}), B (0, 0)$ and $C (2, 0)$ be the vertices
of $∆ A B C$ . Then
$a = B C = 2, b = A C = 2$ and $c = A B = 2$
$\Rightarrow A B = B C = C A$
$\Rightarrow △ A B C$ is equilateral
Hence, the incentre is same as the centroid whose coordinates are
$(\frac{1 + 0 + 2}{3}, \frac{\sqrt{3} + 0 + 0}{3}) \equiv (1, \frac{1}{\sqrt{3}})$

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