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Introduction to 3D coordinate geometry — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSIntroduction to 3D coordinate geometry3 Marks
Question
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
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Answer

Let the triangle formed be $\triangle\text{ABC}$
$\text{(AB)}=\sqrt{(1-2)^2+(2-3)^2+(3-1)^2}$
$=\sqrt{(-1)^2+(-1)^2+(2)^2}$
$=\sqrt{6}\text{ units}$
$\text{BC}=\sqrt{(2-3)^2+(3-1)^2+(1-2)^2}$
$=\sqrt{(-1)^2+(2)^2+(-1)^2}$
$=\sqrt{6}\text{ units}$
$\text{AC}=\sqrt{(1-3)^2+(2-1)^2+(3-2)^2}$
$=\sqrt{(-2)^2+(1)^2+(1)^2}$
$=\sqrt{6}\text{ units}$
since, AB = BC = CA
So, $\triangle\text{ABC}$ is an equilateral $\triangle$

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