Exercises: Distances, parallel and perpendicular lines

Calculate the distance of the line to the origin.

$y = \frac{3}{4} x - 5$

For this task you need the following basic knowledge: Slope/Gradient of a line
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
$m_{2} = - \frac{1}{m_{1}}$
Plug in the known value of $m_{1}$ .
$m_{2} = - \frac{1}{\frac{3}{4}} = - \frac{4}{3}$
So the perpendicular line has the line equation $h (x) : y = - \frac{4}{3} x$ .
Now calculate the intersection point $A (x_{s} | y_{s})$ of the two lines by setting equal their line equations.
$g (x_{s})$ $=$ $h (x_{s})$
↓
Plug in the line equations.
$- \frac{4}{3} x_{s}$ $=$ $\frac{3}{4} x_{s} - 5$ $- \frac{3}{4} x_{s}$
↓
Get the variable $x_{s}$ on the left side
$- \frac{25}{12} x_{s}$ $=$ $- 5$ $\cdot (- 1)$
$\frac{25}{12} x_{s}$ $=$ $5$ $: \frac{25}{12}$
$x_{s}$ $=$ $\frac{12}{5}$
$=$ $2.4$
Now substitute $x_{s}$ into one of the line equations to determine $y_{s}$ .
$h (x_{s}) : y_{s} = - \frac{4}{3} x_{s}$
Plug in $x_{s} = 2.4$ .
$y_{s} = - \frac{4}{3} \cdot 2.4 = - 3.2$
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.
$d = \sqrt{(x_{s} - x_{0})^{2} + (y_{s} - y_{0})^{2}}$
Plug in the values.
$d = \sqrt{(2.4 - 0)^{2} + (- 3.2 - 0)^{2}}$
Simplify.
$= \sqrt{5.76 + 10.24} = \sqrt{16} = 4$
The shortest distance of the line $g$ to the origin is therefore $4$ .
$y = - \frac{1}{2} x + 2$

For this task you need the following basic knowledge: Slope/Gradient of a line
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
$m_{2} = - \frac{1}{m_{1}}$
Plug in the known value of $m_{1}$ .
$m_{2} = - \frac{1}{- \frac{1}{2}} = 2$
So the perpendicular line has the line equation $h (x) : y = 2 x$ .
Now calculate the intersection point $A (x_{s} | y_{s})$ of the two lines by setting equal their line equations.
$g (x_{s})$ $=$ $h (x_{s})$
↓
Plug in the line equations.
$- \frac{1}{2} x_{s} + 2$ $=$ $2 x_{s}$ $- 2 x_{s}$
↓
Get the variable $x_{s}$ on the left side
$- 2.5 x_{s} + 2$ $=$ $0$ $- 2$
↓
Get the 2 to the right side.
$- 2.5 x_{s}$ $=$ $- 2$ $: (- 2.5)$
$x_{s}$ $=$ $0.8$
Now substitute $x_{s}$ into one of the line equations to determine $y_{s}$ .
$h (x_{s}) : y_{s} = 2 x_{s}$
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.
$d = \sqrt{(x_{s} - x_{0})^{2} + (y_{s} - y_{0})^{2}}$
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 $1.79$ .

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$g (x_{s})$	$=$	$h (x_{s})$
	↓	Plug in the line equations.
$- \frac{4}{3} x_{s}$	$=$	$\frac{3}{4} x_{s} - 5$	$- \frac{3}{4} x_{s}$
	↓	Get the variable $x_{s}$ on the left side
$- \frac{25}{12} x_{s}$	$=$	$- 5$	$\cdot (- 1)$
$\frac{25}{12} x_{s}$	$=$	$5$	$: \frac{25}{12}$
$x_{s}$	$=$	$\frac{12}{5}$
	$=$	$2.4$