Question 177791
To find the shortest distance from point H to line JK you need to find a line perpendicular to JK that passes through point H. Find the intersection point between the two lines and then calculate the distance.
First, find the slope of line JK. 
{{{m[JK]=(y[2]-y[1])/(x[2]-x[1])=(-4-4)/(-2-(-6))=-8/4=-2)}}}
Perpendicular lines have slopes that are negative reciprocals of each other.
{{{m[JK]m[P]=-1}}}
{{{-2m[P]=-1}}}
{{{m[P]=1/2}}}
Use the point slope form of the line using this slope and point H.
{{{y-y[0]=m(x-x[0])}}}
{{{y-2=(1/2)(x-5)}}}

{{{y-2=(1/2)x-5/2}}}
{{{y=(1/2)x-1/2}}}
{{{ drawing( 300, 300, -10, 10, -10, 10, grid(1), circle(5,2,0.2),circle(-6,4,0.2),
green(line(-2,-4,-6,4)),
circle(-2,-4,-.2), graph( 300, 300, -10, 10, -10, 10, (1/2)(x-1),-2x-8)) }}}
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Next step is to find the intersection point of the two lines. 
The first line equation is,
{{{y-4=-2(x-(-6))}}}
{{{y-4=-2x-12}}}
{{{y=-2x-8}}}
Set the two equations equal to each other and solve for x, then y.
{{{(1/2)x-1/2=-2x-8}}}
{{{x-1=-4x-16}}}
{{{5x=-15}}}
{{{x=-3}}}
Then
{{{y=-2x-8}}}
{{{y=-2(-3)-8}}}
{{{y=6-8}}}
{{{y=-2}}}
The intersection between the two lines occurs at (-3,-2).
{{{ drawing( 300, 300, -10, 10, -10, 10, grid(1), circle(5,2,0.2),circle(-6,4,0.2),
green(line(-2,-4,-6,4)),
circle(-2,-4,-.2),circle(-3,-2,.3), graph( 300, 300, -10, 10, -10, 10, (1/2)(x-1),-2x-8)) }}}
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Finally, use the distance formula to calculate the distance from H to the intersection point.
{{{D^2=(x[2]-x[1])^2+(y[2]-y[1])^2}}}
{{{D^2=(-3-5)^2+(-2-2)^2}}}
{{{D^2=(-8)^2+(-4)^2}}}
{{{D^2=64+16}}}
{{{D^2=80}}}
{{{D=sqrt(80)}}}