College Algebra, Chapter 2, 2.1, Section 2.1, Problem 42

a.) Show that the points $(7,3)$ and $(3,7)$ are the same distance from the origin.
By using distance formula from origin to point $(7,3)$

$
\begin{equation}
\begin{aligned}
d_1 &= \sqrt{(3-0)^2 + (7-0)^2}\\
\\
&= \sqrt{3^2 + 7^2}\\
\\
&= \sqrt{9+49}\\
\\
&= \sqrt{58} \text{ units}
\end{aligned}
\end{equation}
$

By using distance formula from origin to point $(3,7)$

$
\begin{equation}
\begin{aligned}
d_2 &= \sqrt{(7-0)^2 + (3-0)^2}\\
\\
&= \sqrt{7^2 + 3^2}\\
\\
&= \sqrt{49+9}\\
\\
&= \sqrt{58} \text{ units}
\end{aligned}
\end{equation}
$


It shows that $d_1 = d_2$

b.) Show that the points $(a,b)$ and $(b,a)$ are the same distance from the origin.
By using distance formula from origin to point $(a,b)$

$
\begin{equation}
\begin{aligned}
d_3 &= \sqrt{(b-0)^2 + (a-0)^2}\\
\\
&= \sqrt{b^2 + a^2} \text{ units}
\end{aligned}
\end{equation}
$

By using distance formula from origin to point $(b,a)$

$
\begin{equation}
\begin{aligned}
d_4 &= \sqrt{(a-0)^2 + (b-0)^2}\\
\\
&= \sqrt{a^2 + b^2} \text{ units}
\end{aligned}
\end{equation}
$

It shows that $d_3 = d_4$