Question

Show that the points (1, 2, 3), (2, 3, 1) and (3, 1, 2) form an equilateral triangle.

Answer 1

Step-by-step explanation:

which is required ans.

Answer 2

The given points form an equilateral triangle.

The proof has been shown below.

Step-by-step explanation:

Let the given points are

A(1,2,3), B(2,3,1) and C(3,1,2)

An equilateral triangle has all three sides of equal length.

Hence, for equilateral triangle,

AB = BC = CA

The distance formula is given by

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

Now, find the length of all three sides and check if they are equal or not.

$AB=\sqrt{(2-1)^2+(3-2)^2+(1-3)^2}\\\\AB=\sqrt{1+1+4}\\\\AB=\sqrt{6}$

Now, find the length of side BC

$BC=\sqrt{(3-2)^2+(1-3)^2+(2-1)^2}\\\\BC=\sqrt{1+4+1}\\\\BC=\sqrt{6}$

Finally, find the length of side AC

$AC=\sqrt{(3-1)^2+(1-2)^2+(2-3)^2}\\\\AC=\sqrt{4+1+1}\\\\AC=\sqrt{6}$

Since, the length of all three sides are equal.

Hence, the given points form an equilateral triangle.

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