Question 523748
One diagonal of a rhombus has the same length, 10cm, as each side. How long is the other diagonal


Then it it is two equilateral triangles with a common side, like this,
where all the black line segments are 10 cm each:

{{{drawing(3000/11,400,-.5,2.5,-2.2,2.2, triangle(0,0,2,0,1,sqrt(3)),
triangle(0,0,2,0,1,-sqrt(3)), locate(.1,.9,10cm),


green(line(1,sqrt(3),1,-sqrt(3)))
  )}}}

We want to know the length of the green line segment:

So we look at just one half of the top which is this:

{{{drawing(3000/11,400,-.5,2.5,-2.2,2.2, triangle(0,0,1,0,1,sqrt(3)),
locate(.1,.9,10cm), locate(.5,0,5cm), 
green(line(1,sqrt(3),1,0),locate(1.1,.8,b))
  )}}}

The bottom side is 5cm because the diagonals of a parallelogram bisect
each other (and a rhombus is a parallelogram). The hypotenuse is a side
of the rhombus, so it is 10cm, so the green line in that triangle is
found by the Pythagorean theorem:

  c² = a² + b²

 10² = 5² + b²

 100 = 25 + b²

  75 = b²

{{{sqrt(75)}}} = b 

{{{sqrt(25*3)}}} = b

5{{{sqrt(3)}}} = b

So the entire diagonal is twice that or

10{{{sqrt(3)}}} or about 17.32cm 

Edwin</pre>