Question 1131849
points B({{{b}}},{{{0}}}) and C({{{c}}},{{{10}}})
point A is at ({{{0}}},{{{4}}}) and point D is at ({{{8}}},{{{9}}})

slope of AB is :

{{{m=(4-0)/(0-b)=-4/b}}}

slope of BC is: 

{{{m=(10-0)/(c-b)=10/(c-b)}}}

slope of CD is :

{{{m=(9-10)/(8-c)=-1/(8-c)}}}

The minimum distance  from A to B to C to D will be if the slopes of AB and CD are the same and the slope of BC is the opposite.

The slope of AB is the same as the slope of CD:

{{{-4/b=-1/(8-c)}}}
{{{4c-32=-b}}}
{{{4c=32-b}}}............... (1)

The slope of BC is the opposite of the slope of AB:

{{{10/(c-b)=4/b}}}
{{{4c-4b=10b}}}
{{{4c=14b}}}.............. (2)

from (1) and (2):

{{{32-b=14b}}}
{{{32=14b+b}}}
{{{15b=32}}}
{{{b=32/15}}}
{{{b=2& 2/15}}}