The shortest connection of any point on the line $g(x):y=m_1\cdot x+b$ to the origin is a second line $h(x):y=m_2\cdot x+b_2$ which passes through the origin and is perpendicular to $g$.

Passing through the origin means $h(0)=b_2=0$.

Being perpendicular means

$${\displaystyle m_2=-\frac{1}{m_1}}$$

Plug in the known value of $m_1$.

$${\displaystyle m_2=-\frac{1}{\frac{3}{4}}=-\frac{4}{3}}$$

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

Now calculate the intersection point $A(x_s|y_s)$ of the two lines by setting equal their line equations.

Now substitute $x_s$ into one of the line equations to determine $y_s$.

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

Plug in $x_s=2.4$ .

$${\displaystyle y_s=-\frac{4}{3}\cdot 2.4=-3.2}$$

The intersection of the lines $h$ and $g$ is therefore at $A(2.4|-3.2)$.

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

$${\displaystyle d=\sqrt{(x_s-x_0{)}^2+(y_s-y_0{)}^2}}$$

Plug in the values.

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

Simplify.

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

The shortest distance of the line $g$ to the origin is therefore $4$.

The shortest connection of any point on the line $g(x):y=m_1\cdot x+b$ to the origin is a second line $h(x):y=m_2\cdot x+b_2$ which passes through the origin and is perpendicular to $g$.

Passing through the origin means $h(0)=b_2=0$.

Being perpendicular means

$${\displaystyle m_2=-\frac{1}{m_1}}$$

Plug in the known value of $m_1$.

$${\displaystyle m_2=-\frac{1}{-\frac{1}{2}}=2}$$

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

Now calculate the intersection point $A(x_s|y_s)$ of the two lines by setting equal their line equations.

Now substitute $x_s$ into one of the line equations to determine $y_s$.

$h(x_s):y_s=2x_s$

Plug in $x_s=0.8$ .

$y_s=2\cdot 0.8=1.6$

The intersection of the lines $h$ and $g$ is therefore at $A(0.8|1.6)$.

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

$${\displaystyle d=\sqrt{(x_s-x_0{)}^2+(y_s-y_0{)}^2}}$$

Plug in the values.

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

Simplify.

$=\sqrt{0.64+2.56}=\sqrt{3.2}\approx 1.79$

The shortest distance of the line $g$ to the origin is therefore approximately $1.79$.