For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

Let the length of the dashed line be  x .

 

The sum of the lengths of any two sides in a triangle must be greater than the length of the third side.

 

From the top triangle, we can say...

 

8 + 10  >  x

18  >  x

x  <  18

 

8 + x  >  10

x  >  2

 

10 + x  >  8

x  >  -2

 

From the bottom triangle, we can say....

 

12 + 16  >  x

28  >  x

x  <  28

 

12 + x  >  16

x  >  4

 

16 + x  >  12

x  >  -4

 

So  x  has the restrictions:    x  <  18 ,   x  >  2 ,   x  >  -2 ,   x  <  28 ,   x  >  4 ,   x  >  -4

 

That is....    x  <  18   and  x  >  4

 

So the whole numbers that  x  can be are:

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

 

There are  13  different whole numbers that could be the length of the dashed line.