SOLUTION: four points P,Q,R,S are such that PQ=10,QR=30,RS=15 and PS=m. If m is an integer and no three of these points lie on a straight line, what is the number of possible values of m?

 


Notice in the drawing that QR = QB+BC+CR = 10+5+15 = 30 units, as given.

Let's first of all assume that it would not matter if three of the
points P,Q,R,S could lie on a straight line.  Then  

P could be any point on the circle on the left.
S could be any point on the circle on the right.

Therefore PS is longest when P is at A and S is at D.
In that case the length of PS would be 

AD = AQ+QB+BC+CR+RD = 10+10+5+15+15 = 55 units

and PS is shortest when P is at B and S is at C.
In that case the length of PS would be 

BC = 5 units

Since m is an integer it could be any of these values

5,6,7,...,53,54,55

But when we require that no three of P,Q,R,S can lie on a
straight line, we must rule out 5 and 55, so the possible
values are 

6,7,8,...,52,53,54

There are 54 integers from 1 through 54 inclusive, and 
we cannot use 1,2,3,4 or 5, so there are 54-5 = 49 
possible integer values for m, the length of PS.

Answer: 49

Edwin