If the coordinates of the vertices of an equilateral triangle with sides of length $a$ are $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ , then ${∣ ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣}^{2} =$

\frac{3 a^{4}}{16}

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\frac{4 a^{4}}{3}

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\frac{3 a^{4}}{5}

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\frac{3 a^{3}}{4}

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Solution

The correct option is A $\frac{3 a^{4}}{16}$
We know that
$∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣$
$∴ (A r e a o f e q u i l . Δ)^{2} = {(\frac{\sqrt{3}}{4} a^{2})}^{2}$
$= \frac{3 a^{4}}{16}$

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Similar questions

Q. If

A (x_{1}, y_{1}), B x_{2}, y_{2}), C (x_{3}, y_{3})

are the vertices of an equilateral triangle and

(x_{1} - 2)^{2} + (y_{1} - 3)^{2} = (x_{2} - 2)^{2} + (y_{2} - 3)^{2} = (x_{3} - 2)^{2} + (y_{3} - 3)^{2}

, then

Q. If

A (x_{1}, y_{1}), B (x_{2}, y_{2})

and

C (x_{3}, y_{3})

are vertices of an equilateral triangle whose each side is equal to

a

, the prove that

{⎡ ⎢ ⎣ \begin{matrix} x_{1} & y_{1} & 2 x_{2} & y_{2} & 2 x_{3} & y_{3} & 2 \end{matrix} ⎤ ⎥ ⎦}^{2} = {3 a}^{4}

Q. If

A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})

are the vertices of an equilateral triangle and

(x_{1} - 4)^{2} + (y_{1} - 5)^{2} = (x_{2} - 4)^{2} + (y_{2} - 5)^{2} = (x_{3} - 4)^{2} + (y_{3} - 5)^{2},

then the value of

(y_{1} + y_{2} + y_{3}) - (x_{1} + x_{2} + x_{3})

If the coordinates of the vertices of an equilateral triangle with sides of length 'a' are $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ , then
${∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣}^{2} = \frac{3 a^{4}}{4}$

Find the centroid of the triangle with vertices A $(x_{1}, y_{1})$ , B $(x_{2}, y_{2})$ and C $(x_{3}, y_{3})$ .

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If the coordinates of the vertices of an equilateral triangle with sides of length a are (x1,y1),(x2,y2) and (x3,y3), then ∣∣ ∣ ∣∣x1y11x2y21x3y31∣∣ ∣ ∣∣2=

The correct option is A 3a416We know that∣∣ ∣∣x1y11x2y21x3y31∣∣ ∣∣∴(Areaofequil.Δ)2=(√34a2)2=3a416

If the coordinates of the vertices of an equilateral triangle with sides of length $a$ are $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ , then ${∣ ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣}^{2} =$

The correct option is A $\frac{3 a^{4}}{16}$
We know that
$∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣$
$∴ (A r e a o f e q u i l . Δ)^{2} = {(\frac{\sqrt{3}}{4} a^{2})}^{2}$
$= \frac{3 a^{4}}{16}$