Question 570227
They want the SHORTest distance from the
point to the line. There are lots of lines you
could draw from the point to the line, but
only 1 is the shortest.
That shortest distance is the perpendicular
from the point to the line.
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The 1st thing to do is find the slope of the line.
Get it into the form {{{ y = mx + b }}} where 
{{{ m }}} is the slope. ( It's already in that form )
{{{ y = x + 1 }}} The slope is {{{ 1 }}}.
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Now you want the slope of ANY line which would
be perpendicular to this line.
That formula is {{{ -( 1/m ) = -(1/1) }}},
so the perpendicular will have slope = {{{ -1 }}}
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Now you have the point (-2,3) and a slope. Use
the point-slope formula to get the equation of
the perpendicular line.
{{{ ( y - 3 ) / ( x -(-2) ) = -1 }}}
{{{ y - 3 = -1*( x + 2 ) }}}
{{{ y = -x - 2 + 3 }}}
{{{ y = -x + 1 }}}
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Now where do these 2 lines intersect?
Add the equations:
{{{ y = x + 1 }}}
{{{ y = -x + 1 }}}
{{{ 2y = 2 }}}
{{{ y = 1 }}}
and
{{{ y = x + 1 }}}
{{{ 1 = x + 1 }}}
{{{ x = 0 }}}
So now you have 2 points, (-2,3) and (0,1)
The formula for distance is
distance = {{{ sqrt( ( -2 - 0)^2 + ( 3 - 1 )^2 ) }}}
distance = {{{ sqrt( (-2)^2 + 2^2 ) }}}
distance = {{{ sqrt( 4 + 4 ) }}}
distance = {{{ 2*sqrt(2) }}}
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Here's a plot of the line {{{ y = x + 1 }}} and the line
perpendicular to it going through (-2,3)
The intersection is at (0,1) as you can see
{{{ graph( 400, 400, -5, 5, -5, 5, x + 1, -x + 1) }}}