Question 1058769
Line AB is equation {{{y-5=((8-5)/(13-7))(x-7)}}}
{{{y=5+(1/2)(x-7)}}}
{{{y=x/2-7/2+10/2}}}
{{{y=x/2+3/2}}}



Distance from A to B is {{{sqrt((13-7)^2+(8-5)^2)}}}
{{{sqrt(36+9)}}}
{{{sqrt(45)}}}
{{{3sqrt(5)}}}


You want to know the point on the line so that 
{{{(1/3)(3sqrt(5))=sqrt((x-7)^2+(y-5)^2)}}}
and you want to substitute for y and then solve this equation for x.


{{{sqrt(5)=sqrt((x-7)^2+(x/2+3/2-5)^2)}}}


{{{sqrt(5)=sqrt((x-7)^2+(x/2+3/2-10/2)^2)}}}


{{{sqrt(5)=sqrt((x-7)^2+(x/2-7/2)^2)}}}
square both sides.
{{{5=(x-7)^2+(x/2-7/2)^2}}}
{{{5=x^2-14x+49+(1/4)(x-7)^2}}}
{{{20=4x^2-56x+196+x^2-14x+49}}}
{{{20=5x^2-70x+245}}}
{{{5x^2-70x+220=0}}}
{{{highlight_green(x^2-14x+44=0)}}}-------not factorable so use general solution for quadratic formula.


{{{x=(14+- sqrt(14^2-4*44))/2}}}


{{{x=(14+- sqrt(108))/2}}}


{{{x=(14+- sqrt(9*4*3))/2}}}


{{{x=(14+- 12*sqrt(3))/2}}}


{{{highlight(x=7+- 6*sqrt(3))}}}-------Not finished.  One of these will work and the other will not work.  Most likely the PLUS form will be the correct choice; and then find the corresponding y value from the equation of the line AB.