Q.If the co-ordinates of the vertices of an equilateral triangle with sides of length $a$ are $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, then $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}^2 = \dfrac{3a^4}{4}$.

The key idea is that the absolute value of the determinant gives twice the area of the triangle. For an equilateral triangle of side $a$, the area is $\frac{\sqrt{3}}{4}a^2$.

Step 1: The area $\Delta$ of any triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ is

Step 2: For an equilateral triangle of side $a$, the area is $\Delta = \frac{\sqrt{3}}{4}a^2$.

Step 3: Equate the two expressions: …

The key insight here is that the determinant

has a direct geometric meaning: its absolute value equals twice the area of the triangle formed by the three points. This is a standard result from coordinate geometry — the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is $\frac{1}{2}|\Delta|$.

So the problem reduces to: for an equilateral triangle of side $a$, what is its area? Square that, multiply by 4, and you have the answer.

Let's work through it step by step.

1. Area of an equilateral triangle For any triangle, area = $\frac{1}{2} \times \text{base} \times \text{height}$. In an equilateral triangle of side $a$, the height is $\frac{\sqrt{3}}{2}a$ (by Pythagoras: $h^2 = a^2 - (a/2)^2$). So the area is:

2. Relating the determinant to area As noted, $|\Delta| = 2 \times \text{Area}$. Therefore:

3. Squaring the determinant …

Method: Linking the Coordinate-Determinant Formula to a Known Area

This method is for questions that give a geometric shape (like an equilateral triangle) and ask you to relate its side length to the determinant built from its vertices' coordinates — the bridge is the standard area-by-determinant formula.

Steps

Step 1: Write the determinant-area relationship

For any triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, the signed area is half the determinant:

Step 2: Bring in the shape-specific area formula

Use whatever formula the problem's shape supplies — for an equilateral triangle of side $a$, that is $\text{Area}=\frac{\sqrt3}{4}a^2$. Substitute this into the relation from Step 1 to express $D$ purely in terms of the given side/measurement.

Step 3: Match the power asked for in the question …

Common Mistakes

Mistake 1: Confusing the determinant with the area directly

The determinant is twice the (signed) area, $|\Delta|=2\times\text{Area}$ — not equal to the area. Forgetting the factor of $2$ before squaring throws the final answer off by a factor of $4$.

Mistake 2: Using the wrong height for an equilateral triangle

The height of an equilateral triangle of side $a$ is $h=\frac{\sqrt3}{2}a$, derived from $h^2=a^2-(a/2)^2$. Misremembering this as $h=\frac{a}{2}$ or $h=\sqrt3\,a$ gives a wrong area and a wrong final value.

Mistake 3: Worrying about the sign of $\Delta$ unnecessarily …