PROBLEM SOLVING QUESTIOn!!!?

Question

How many different triangles all with whole number lengths have a perimeter of 24cm?

Answer 1

12 11 1

11 11 2

10 11 3

9  11 4

8   11 5

7   11  6

6   11  7

5    11 8

4     11 9

3      11  10

2     11   11

1         12

Answer 2

8x8x8

Answer 3

Infinitely many because if you take one side's length as 1cm then other 2 can be 22&1 or 21&2 or 20&3 or...........

Answer 4

The longest side must be less than half the perimiter.  A triangle with sides 12, 11, 1 would be a straight line.  There can be no 1,1,22.

8,8,8 is only one of many choices.

Let your longest side be 11.  Then the other sides must sum to 13.

11,11,2 /\ 11,10,3 /\  11,9,4 /\ 11,8,5 /\  11,7,6  /\ 11,6,7  /\ 11,5,8 /\ 11,4,9 /\ 11,3,10 /\ 11,2,11   10 triangles.

Now let the longest side be 10 and the other sides sum to 14.

10,11,3  /\ 10,10,4 /\ 10,9,5  /\ 10,8,6 /\ 10,7,7 /\ 10,6,8  10,5,9 /\ 10,4,10 /\ 10,3,11  But the 11,3 combinations have already been used above.  

7 triangles.

Longest side 9, Other sides sum to 15.

9,11,4 /\  9,10,5 /\  9,9,6 /\  9,8,7 /\ 9,7,8 /\  9,6,9 /\ 9,5,10 /\  9,4,11.

the 11,4 anf 10,5 combinations have been used.  4 triangles.

longest side 8.  Other sides sum to 16

8,11,5 /\  8,10,6 /\ 8,9,7 /\ 8,8,8 /\ 8,7,9 /\ 8,6,10 /\ 8,5,11

11,5  10,6  amd 9,7  combinations have been used.  1 triangle.

10+7+4+1 = 22