Question 1203482
 the three vertices of a parallelogram are
 
({{{ 1}}},{{{ 3}}})
({{{ 0}}},{{{ 0}}})
({{{ 4}}},{{{ 0}}}) 


Find the three possible locations of the fourth vortex

a parallelogram is a figure with opposite sides parallel

the fourth vortex is ({{{ x}}},{{{  y}}})

line that passes through ( {{{ 0}}},{{{ 0}}}) and ({{{ 1}}},{{{ 3}}}) is parallel to line that passes through ({{{ 4}}},{{{ 0}}}) and ({{{  x}}},{{{ y}}})

so, find a slope of the line that passes through ({{{ 0}}},{{{ 0}}}) and ({{{ 1}}},{{{ 3}}}) :

 {{{ m= (3-0)/(1-0)=3}}}


parallel lines have same slope, so  line that passes through ({{{ 4,{{{ 0}}}) and ( {{{ x}}},{{{  y}}}) will be

{{{ y-y1=3(x-x1)}}}..use point ({{{ 4}}},{{{ 0}}})

{{{ y-0=3(x-4)}}}

{{{ y=3x-12}}}

now, we know that point  ({{{x}}},{{{y}}}) will be intersection of tis line and a line that passes through ({{{1}}},{{{3}}}) and is parallel to x-axis, or a line {{{y=3}}}


{{{ 3x-12=3}}}
{{{ 3x=3+12}}}
{{{ x=15/3}}}
{{{ x=5}}}

and

{{{ y=3*5-12}}}
{{{ y=3}}}

 ({{{x}}},{{{y}}}) = ({{{ 5}}},{{{ 3}}}) 



{{{ drawing( 600, 600, -10, 10, -10, 10, 
green(line(0,0,1,3)),green(line(4,0,5,3)),green(line(1,3,5,3)),
locate(5,3,p(5,3)),
graph( 600, 600, -10, 10, -10, 10, 0)) }}}