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If the extremities of a diagonal of a square are $(1,2,3)$ and $(2,-3,5)$, then its side is of length
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2912 Upvotes
Verified Answer
The correct answer is:
$\sqrt{15}$
Let the side of the square be a.
Then,
$$
\begin{aligned}
A C & =\sqrt{(2-1)^2+(-3-2)^2+(5-3)^2} \\
& =\sqrt{1+25+4}=\sqrt{30}
\end{aligned}
$$
Now, $\quad A B^2+B C^2=A C^2$
$$
\begin{aligned}
\Rightarrow & a^2+a^2 & =30 \Rightarrow 2 a^2=30 \\
\Rightarrow & a^2 & =15 \Rightarrow a=\sqrt{15} \text { units }
\end{aligned}
$$
Then,
$$
\begin{aligned}
A C & =\sqrt{(2-1)^2+(-3-2)^2+(5-3)^2} \\
& =\sqrt{1+25+4}=\sqrt{30}
\end{aligned}
$$
Now, $\quad A B^2+B C^2=A C^2$
$$
\begin{aligned}
\Rightarrow & a^2+a^2 & =30 \Rightarrow 2 a^2=30 \\
\Rightarrow & a^2 & =15 \Rightarrow a=\sqrt{15} \text { units }
\end{aligned}
$$
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