The shortest connection of any point on the line $g(x):y=m_1\cdot x+bg(x):y=m\_1\backslash cdot\; x+b$ to the origin is a second line $h(x):y=m_2\cdot x+b_{2}h(x):\; y\; =\; m\_2\; \backslash cdot\; x\; +\; b\_2$ which passes through the origin and is perpendicular to $gg$.

Passing through the origin means $h(0)=b_2=0h(0)\; =\; b\_2\; =\; 0$.

Being perpendicular means

${\displaystyle m_2=-\frac{1}{m_1}}\backslash displaystyle\; m\_2=-\backslash frac1\{m\_1\}$

Plug in the known value of $m_{1}m\_1$.

${\displaystyle m_2=-\frac{1}{\frac{3}{4}}=-\frac{4}{3}}\backslash displaystyle\; m\_2=-\backslash frac\{1\}\{\backslash frac34\}=-\backslash frac43$

So the perpendicular line has the line equation ${\displaystyle h(x):y=-\frac{4}{3}x}\backslash displaystyle\; h(x):y=-\backslash frac43\; x$.

Now calculate the intersection point $A(x_s\mathrm{\mid }{y}_s)A\; \backslash ,(x\_s|y\_s)$ of the two lines by setting equal their line equations.

Now substitute $x_{s}x\_s$ into one of the line equations to determine $y_{s}y\_s$.

${\displaystyle h(x_s):y_s=-\frac{4}{3}{x}_s}\backslash displaystyle\; h(x\_s):y\_s=-\backslash frac43x\_s$

Plug in $x_s=2.4x\_s=2.4$ .

${\displaystyle y_s=-\frac{4}{3}\cdot 2.4=-3.2}\backslash displaystyle\; y\_s=-\backslash frac43\backslash cdot2.4=-3.2$

The intersection of the lines $hh$ and $gg$ is therefore at $A(2.4\mathrm{\mid }-3.2)A\backslash ,(2.4|-3.2)$.

Now determine the distance of the origin to the calculated intersection point $AA$, this is exactly the shortest distance of the line $gg$ to the origin.

${\displaystyle d=\sqrt{(x_s-x_0{)}^2+(y_s-y_0{)}^2}}\backslash displaystyle\; d=\backslash sqrt\{(x\_s-x\_0)^2+(y\_s-y\_0)^2\}$

Plug in the values.

${\displaystyle d=\sqrt{(2.4-0{)}^2+(-3.2-0{)}^2}}\backslash displaystyle\; d=\backslash sqrt\{(2.4-0)^2+(-3.2-0)^2\}$

Simplify.

${\displaystyle =\sqrt{5.76+10.24}=\sqrt{16}=4}\backslash displaystyle\; =\backslash sqrt\{5.76+10.24\}=\backslash sqrt\{16\}=4$

The shortest distance of the line $gg$ to the origin is therefore $44$.

The shortest connection of any point on the line $g(x):y=m_1\cdot x+bg(x):y=m\_1\backslash cdot\; x+b$ to the origin is a second line $h(x):y=m_2\cdot x+b_{2}h(x):\; y\; =\; m\_2\; \backslash cdot\; x\; +\; b\_2$ which passes through the origin and is perpendicular to $gg$.

Passing through the origin means $h(0)=b_2=0h(0)\; =\; b\_2\; =\; 0$.

Being perpendicular means

${\displaystyle m_2=-\frac{1}{m_1}}\backslash displaystyle\; m\_2=-\backslash frac1\{m\_1\}$

Plug in the known value of $m_{1}m\_1$.

${\displaystyle m_2=-\frac{1}{-\frac{1}{2}}=2}\backslash displaystyle\; m\_2=-\backslash frac\{1\}\{-\backslash frac12\}=2$

So the perpendicular line has the line equation $h(x):y=2xh(x):y=2x$.

Now calculate the intersection point $A(x_s\mathrm{\mid }{y}_s)A\; \backslash ,(x\_s|y\_s)$ of the two lines by setting equal their line equations.

Now substitute $x_{s}x\_s$ into one of the line equations to determine $y_{s}y\_s$.

$h(x_s):y_s=2x_{s}h(x\_s):y\_s=2x\_s$

Plug in $x_s=0.8x\_s=0.8$ .

$y_s=2\cdot 0.8=1.6y\_s=2\backslash cdot0.8=1.6$

The intersection of the lines $hh$ and $gg$ is therefore at $A(0.8\mathrm{\mid }1.6)A\backslash ,(0.8|1.6)$.

Now determine the distance of the origin to the calculated intersection point $AA$, this is exactly the shortest distance of the line $gg$ to the origin.

${\displaystyle d=\sqrt{(x_s-x_0{)}^2+(y_s-y_0{)}^2}}\backslash displaystyle\; d=\backslash sqrt\{(x\_s-x\_0)^2+(y\_s-y\_0)^2\}$

Plug in the values.

$d=\sqrt{(0.8-0{)}^2+(1.6-0{)}^2}d=\backslash sqrt\{(0.8-0)^2+(1.6-0)^2\}$

Simplify.

$=\sqrt{0.64+2.56}=\sqrt{3.2}\approx 1.79=\backslash sqrt\{0.64+2.56\}=\backslash sqrt\{3.2\}\backslash approx1.79$

The shortest distance of the line $gg$ to the origin is therefore approximately $1.791.79$.