NIMCET 2013 — Mathematics PYQ

Solution

Concept:

In an equilateral triangle, the perpendicular distance from a vertex to the opposite side (the base) is the altitude ($h$). The relationship between the side length ($a$) and the altitude ($h$) is given by:

Step 1: Find the altitude ($h$).

The altitude is the perpendicular distance from the vertex $(x_1,y_1)=(2,-1)$ to the line $x+y-2=0$.

The formula for the distance $d$ from a point $(x_1,y_1)$ to a line $Ax+By+C=0$ is:

Applying the values:

Step 2: Calculate the side length ($a$).

Substitute the value of $h$ into the side-altitude relation:

Step 3: Simplify the expression.

We can write $2$ as $\sqrt{2}\times \sqrt{2}$:

Final Answer:

The length of the side of the triangle is $\sqrt{\frac{2}{3}}$, which corresponds to Option 3.