Question 535938
Quadrilateral NORA has vertices N(3,2) O(7,0), R(11,2) an A(7,4). Use coordinate geometry to prove that quadrilateral NORA is a rhombus.

{{{drawing(400,220,-5,15,-5,6, graph(400,220,-5,15,-5,6),
circle(3,2,.2), circle(7,0,.2), circle(11,2,.2), circle(7,4,.2),
locate(3.5,2.5,"N(3,2)"), locate(7.5,1,"O(7,0)"), locate(11.2,2.5,"R(11,2)"), locate(7.5,4.5,"A(7,4)"), 

green(line(3,2,7,0), line(7,0,11,2), line(11,2,7,4), line(7,4,3,2))



 )}}}  

We prove that all 4 sides are the same length, using the distance formula:

d = {{{sqrt((x[2]-x[1])^2+(y[2]-y[1])^2)}}}

NO = {{{sqrt((7-3)^2+(0-2)^2)}}} = {{{sqrt(4^2+(-2)^2)}}} = {{{sqrt(16+4)}}} = {{{sqrt(20)}}} = {{{sqrt(4*5)}}} = {{{2sqrt(5)}}}

OR = {{{sqrt((2-0)^2+(11-7)^2)}}} = {{{sqrt(2^2+4^2)}}} = {{{sqrt(4+16)}}} = {{{sqrt(20)}}} = {{{sqrt(4*5)}}} = {{{2sqrt(5)}}}

RA = {{{sqrt((7-11)^2+(4-2)^2)}}} = {{{sqrt((-4)^2+2^2)}}} = {{{sqrt(16+4)}}} = {{{sqrt(20)}}} = {{{sqrt(4*5)}}} = {{{2sqrt(5)}}}

AN = {{{sqrt((7-3)^2+(4-2)^2)}}} = {{{sqrt(4^2+2^2)}}} = {{{sqrt(16+4)}}} = {{{sqrt(20)}}} = {{{sqrt(4*5)}}} = {{{2sqrt(5)}}}

All four sides are of equal length {{{2sqrt(5)}}}

Edwin</pre>