MAH-CET 2026 — Mathematics PYQ

Step 1: Identify the vertices of the triangle

Let the vertices of the triangle be:

• $A=(1,\sqrt{3})$

• $B=(0,0)$

• $C=(2,0)$

Step 2: Find the side lengths of the triangle

We use the distance formula, $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, to determine the lengths of the sides opposite to each vertex:

• Length of side $a$ (opposite to vertex $A$, between $B$ and $C$):

  $$a=\sqrt{(2-0{)}^2+(0-0{)}^2}=\sqrt{4}=2$$

• Length of side $b$ (opposite to vertex $B$, between $A$ and $C$):

  $$b=\sqrt{(2-1{)}^2+(0-\sqrt{3}{)}^2}=\sqrt{1+3}=\sqrt{4}=2$$

• Length of side $c$ (opposite to vertex $C$, between $A$ and $B$):

  $$c=\sqrt{(1-0{)}^2+(\sqrt{3}-0{)}^2}=\sqrt{1+3}=\sqrt{4}=2$$

Since all three sides are equal ($a=b=c=2$), $△ABC$ is an equilateral triangle.

Step 3: Calculate the Incenter

Key Mathematical Property: In any equilateral triangle, the incenter, centroid, circumcenter, and orthocenter all coincide at the exact same point.

Because it is an equilateral triangle, we can simply find the centroid to get the coordinates of the incenter. The formula for the centroid $(X,Y)$ is:

$$X=\frac{x_1+x_2+x_3}{3}$$

$$Y=\frac{y_1+y_2+y_3}{3}$$

Substituting our vertex values:

$$X=\frac{1+0+2}{3}=\frac{3}{3}=1$$

$$Y=\frac{\sqrt{3}+0+0}{3}=\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}$$

Therefore, the coordinates of the incenter are $\left(1,\frac{1}{\sqrt{3}}\right)$.

Conclusion

The incenter of the given triangle is $\left(1,\frac{1}{\sqrt{3}}\right)$.

Correct Option: (d) $\left(1,\frac{1}{\sqrt{3}}\right)$