Question 830754
Part-way through the solution,


The line MN :
 m=(6-0)/(3-9)=-1.
if y=mx+b,then b=y-mx.
pick either point to get b.
b=0-(-1)9
b=9.
The line MN is {{{y=-x+9}}}.


Now, you want some point ON THE LINE MN, y=-x+9, which is 2/3 of the length distance from M(3,6).  Use the distance formula, to first find the distance MN.


{{{MN = sqrt((0-6)^2+(9-3)^2)}}}
{{{sqrt(36+36)}}}
{{{sqrt(72)}}}
{{{sqrt(2*36)}}}
{{{MN=highlight_green(6*sqrt(2))}}}
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{{{(2/3)*(MN)=2*sqrt(2)}}}-------Use this as a target value, and use distance formula again; but now, you have some unknown point, (x,-x+9) and the known point, (3,6), and you want the distance between them to be {{{2*sqrt(2)}}}.


If you understand all of that clearly, then you know what to do.  Note, the variable point to be found, involves the line MN; the x coordinate as unknown, and the resulting y coordinate as -x+9.  
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You will get two solutions for x and one of them will be what you want but the other will not be.