Question 1006745
Prove that quadrilateral {{{PLUS}}} with vertices P({{{2}}},{{{1}}}), L({{{6}}},{{{3}}}), U({{{5}}},{{{5}}}), and S({{{1}}},{{{3}}}) is a rectangle but not a square


by definition,a rectangle is a quadrilateral that has opposite sides are parallel and of equal length 

so, we need to find the distance between vertices which is the length of the sides

if the distance between points {{{P}}} and {{{L}}} is same as the distance between points {{{U}}} and {{{S}}}, then the length of the sides {{{PL}}} and {{{US}}} are same, or {{{PL=US}}}

and if the distance between points {{{U}}} and {{{L}}} is same as the distance between points {{{P}}} and {{{S}}}, then the length of the sides {{{UL}}} and {{{PS}}} are same, or {{{UL=PS}}}

finally, if {{{PL<>PS}}} and {{{US<>UL}}}then a quadrilateral is a rectangle

{{{PL}}} 

*[invoke Distance_Formula_for_Coordinate_Plane 2, 1, 6, 3] 

and {{{US}}}

*[invoke Distance_Formula_for_Coordinate_Plane 5, 5, 1, 3] 


{{{UL}}} 
*[invoke Distance_Formula_for_Coordinate_Plane 5, 5, 6, 3] 
and 

{{{PS}}}

*[invoke Distance_Formula_for_Coordinate_Plane 2, 1, 1, 3] 

as you can see, {{{PL=US}}} and {{{UL=PS}}};so, a quadrilateral is a rectangle


{{{drawing( 600, 600, -10, 10, -10, 10,
circle(2,1,.12),circle(6,3,.12),circle(5,5,.12),circle(1,3,.12),
locate(2,1,P),locate(6,3,L),locate(5,5,U),locate(1,3,S),
line(2,1,1,3),line(5,5,1,3),line(6,3,2,1),line(5,5,6,3),
 graph( 600, 600, -10, 10, -10, 10, 0)) }}}