Question 972911
{{{drawing(300,240,-5,5,-4,4,
line(-4,0,0,3),line(-4,0,0,-3),
line(4,0,0,3),line(4,0,0,-3),rectangle(0,0,0.5,0.5),
green(line(-4,0,4,0)),red(line(0,-3,0,3)),
locate(-2.2,0,green(4)),locate(1.8,0,green(4)),
locate(0.1,1.8,red(3)),locate(0.1,-1.2,red(3)),
locate(1.8,1.5,x)
)}}} The diagonals divide the rhombus into 4 congruent right triangles.
Each side of the rhombus is the hypotenuse of one of those triangles.
Each leg of one of those triangles is half of one diagonal,
so their measures are
{{{green(8cm)/2=green(4cm)}}} and {{{red(6cm)/2=red(3cm)}}} .
According to the Pythagorean theorem, The lengths of the sides of those triangles, are related by
{{{x^2=green(4cm)^2+red(3cm)^2}}} , so
{{{x^2=green(16)cm^2+red(9)cm^2}}}
{{{x^2=25cm^2}}}
{{{x^2=(5cm^2)}}}
{{{highlight(x=5cm)}}}