Question 1123955
The area of the rhombus is {{{540cm^2}}}; the length of one of its diagonals is {{{d[1]=4.5dm=45cm}}}. What is the distance between the point of intersection of the diagonals and the side of the rhombus? 

 

Here is the rhombus with its diagonals.
I put the rhombus inside a rectangular box for safekeeping,
because it looked breakable. 
(I know it is not drawn to scale).
{{{drawing(300,400,-5.5,9.5,-3,3,
triangle(0,0,5,0,3.7,0.624),locate(2.6,0.43,f),
triangle(-5,0,5,0,0,2.4),triangle(0,-2.4,5,0,0,2.4),
line(-5,-2.4,5,-2.4),line(-5,2.4,5,2.4),
line(-5,-2.4,-5,2.4),line(5,-2.4,5,2.4),
red(line(-5,0,0,-2.4)),red(line(5,0,0,-2.4)),
red(line(-5,0,0,2.4)),red(line(5,0,0,2.4)),
locate(5.1,0.1,4.5dm=45cm),arrow(5.5,0.2,5.5,2.4),
arrow(5.5,-0.2,5.5,-2.4),locate(-2.55,0,x),locate(2.45,0,x)
)}}}

You can see that the diagonals split the rhombus into4 congruent right triangles.
Maybe you want to know the distance {{{x}}} (in cm) between the point of intersection of the diagonals and the end of the other diagonal (which is half of the length of the other diagonal).
It is useful to calculate that length, anyway.
The area of a rhombus is the length of one diagonal times half the length of the other.
So, {{{45*x=540}}} ---> {{{x=12}}} .
 
Maybe you really wanted the distance between the point of intersection of the diagonals and the side of the rhombus, measured along the shortest path, the line perpendicular to the red side.
That is {{{f}}} ,the altitude to the hypotenuse of one of those right triangles.
The length {{{f}}} of such an altitude in a right triangle with leg lengths {{{x}}} and half of the other diagonal which is{{{d[1]/2=45/2}}} can be found from

{{{1/f^2=1/x^2+1/(d[1]/2)^2}}} .

{{{1/f^2=1/12^2+1/22.5^2}}}

{{{1/f^2=1/144+1/506.25}}}

{{{1/f^2=(506.25+144)/72900}}}

{{{1/f^2=650.25/72900}}}

=> {{{f^2=72900/650.25}}}

 {{{f=sqrt(72900 /650.25)}}}

{{{f=10.588}}} --> {{{highlight(f=10.59)}}} 


In other words, the distance from the center of the rhombus to one of its sides is approximately {{{highlight(10.59cm)}}} .