SOLUTION: A rhombus has sides of length 7 cm. One of its diagonals is 10 cm long. Find the lengths of the other diagonal.

As you can see from the drawing, it looks almost like 
the rhombus is a square, so the other diagonal should 
be close to 10 also. Let's suppose the horizonral 
diagonal is the one that is 10 cm. Let's draw it in:

   

We want to find the other diagonal, so let's draw it in:

 

Notice that this vertical diagonal divides the 10 cm diagonal
into two equal segments which are 5 cm each. In fact the two
diagonals have divided the rhombus into four congruent right
triangles:



Now we pick only one of the four congruent right 
triangles, say, we pick the upper left one:



The hypotenuse, call it c, is 7, and the horizontal
(bottom) leg a is 5, and we need to find the vertical
side (the right side), which we call b:



so now we use the Pythagorean theorem to find the 
leg b (the right side).

      c² = a² + b²

      7² = 5² + b²

      49 = 25 + b²
Subtract 25 from both sides:

 49 - 25 = b²

      24 = b²
      __
     Ö24 = b
                          _
(This can be written as 2Ö6 if you have
 learned how to get radicals to lowest terms.)

Or it can be approximated with the decimal
       b = 4.898979486 cm.

But the diagonal of the rhombus is twice this
or 
  __      _
2Ö24 or 4Ö6 or 9.797958971 cm.  So this is very
close to 10 cm, so now we know why it looks so
much like a square.  It is very nearly a square.

Edwin