Find the slope of any line perpendicular to the line passing through (24,−2) and (18,19)

Hunter Shah
2022-10-16
Answered

Find the slope of any line perpendicular to the line passing through (24,−2) and (18,19)

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Besagnoe9

Answered 2022-10-17
Author has **9** answers

The first step is to calculate the gradient (m) of the line joining the 2 points using the gradient formula

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

where $({x}_{1},{y}_{1})\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}({x}_{2},{y}_{2})\phantom{\rule{1ex}{0ex}}\text{are the coords of 2 points}\phantom{\rule{1ex}{0ex}}$

let $({x}_{1},{y}_{1})=(24,-2)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}({x}_{2},{y}_{2})=(18,19)$

substitute these values into formula for m.

$\Rightarrow m=\frac{19+2}{18-24}=\frac{21}{-6}=-\frac{7}{2}$

Now if 2 lines with gradients $m}_{1}\text{}\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}{m}_{2$ are perpendicular

then their product ${m}_{1}.{m}_{2}=-1$

let $m}_{2}\phantom{\rule{1ex}{0ex}}\text{be gradient of perpendicular line}\phantom{\rule{1ex}{0ex}$

$\Rightarrow {m}_{2}=\frac{-1}{{m}_{1}}=-\frac{1}{-\frac{7}{2}}=\frac{2}{7}$

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

where $({x}_{1},{y}_{1})\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}({x}_{2},{y}_{2})\phantom{\rule{1ex}{0ex}}\text{are the coords of 2 points}\phantom{\rule{1ex}{0ex}}$

let $({x}_{1},{y}_{1})=(24,-2)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}({x}_{2},{y}_{2})=(18,19)$

substitute these values into formula for m.

$\Rightarrow m=\frac{19+2}{18-24}=\frac{21}{-6}=-\frac{7}{2}$

Now if 2 lines with gradients $m}_{1}\text{}\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}{m}_{2$ are perpendicular

then their product ${m}_{1}.{m}_{2}=-1$

let $m}_{2}\phantom{\rule{1ex}{0ex}}\text{be gradient of perpendicular line}\phantom{\rule{1ex}{0ex}$

$\Rightarrow {m}_{2}=\frac{-1}{{m}_{1}}=-\frac{1}{-\frac{7}{2}}=\frac{2}{7}$

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