Question 1165801
The quadrilateral{{{ PQRS}}} has vertices 
{{{P}}}({{{2}}},{{{4}}}) 
{{{Q}}}({{{5}}},{{{1}}}) 
{{{R}}}({{{-1}}},{{{-2}}}) 
{{{S}}}({{{-4}}},{{{1}}})
 

 the opposite sides are:
{{{PQ}}} and {{{RS}}}
{{{PS}}} and {{{QR}}}

use given data to find distance between points or the length of the sides:

{{{P}}}({{{2}}},{{{4}}}) 
{{{Q}}}({{{5}}},{{{1}}})

*[invoke formula_distance 2, 4, 5, 1]

{{{PQ=4.24264068711928}}}


{{{R}}}({{{-1}}},{{{-2}}}) 
{{{S}}}({{{-4}}},{{{1}}})

*[invoke formula_distance -1, -2, -4, 1] 

{{{RS=4.24264068711928}}}

so, {{{PQ=RS}}}

if so, then {{{PS}}} and {{{QR}}} must be equal too

let's check it

{{{P}}}({{{2}}},{{{4}}}) 
{{{S}}}({{{-4}}},{{{1}}})

*[invoke formula_distance 2, 4, -4, 1]

{{{PS=6.70820393249937}}}

{{{Q}}}({{{5}}},{{{1}}})
{{{R}}}({{{-1}}},{{{-2}}}) 

*[invoke formula_distance 5, 1, -1, -2]

{{{QR=6.70820393249937}}}

and, {{{PS = QR}}}

it is proven that {{{PQRS }}}is a parallelogram  


{{{ drawing( 600, 600, -10, 10, -10, 10, 
circle(5,1,.12),circle(-1,-2,.12),
circle(2,4,.12),circle(-4,1,.12),
green(line(5,1,-1,-2)),green(line(2,4,-4,1)),
green(line(-1,-2,-4,1)),green(line(2,4,5,1)),
 graph( 600, 600, -10, 10,-10,10, 0)) }}}