Question 1009558
find the coordinates of the point {{{P}}} that lies along the directed line segment from  A({{{3}}},{{{4}}}) TO B({{{6}}},{{{10}}}) and partitions the segment in the ratio {{{3: 2}}}

 To find the point {{{P}}} that divides a segment {{{AB}}} into a particular ratio, determine the ratio {{{k}}} by writing the numerator over the sum of the numerator and the denominator of the given ratio. Next, find the rise and the run (slope) of the line. Finally, add {{{k* the_run}}} to the x-coordinate of {{{A}}} and add {{{k* the _rise}}} to the y-coordinate of {{{A}}}. This process is summarized with the following formula.


({{{x}}},{{{y}}})=({{{x[1]+k(x[2]-x[1])}}},{{{y[1]+k(y[2]-y[1])}}})

so,
the ratio {{{3: 2}}}=> partitions the segment into {{{5}}} congruent pieces

the slope of {{{AB}}} is {{{run/rise=(10-4)/(6-3)=6/3}}}

find the coordinates of the point {{{P}}} , 
add {{{3/5}}} of the run to the x-coordinate of point {{{A}}}
and {{{3/5}}} of the rise to the y-coordinate of point {{{A}}}

run: {{{(3/5)3=1.8}}}  
rise:{{{( 3/5) 6=3.6}}}

so,  the coordinates of the point {{{P}}} are:
 
({{{3+1.8}}},{{{4+3.6}}})

({{{4.8}}},{{{7.6}}})

the ratio of {{{AP:PB}}} is {{{3:2}}}