Question 217905: The lengths of the sides of a triangle are all whole numbers. The triangle has a perimeter (distance around the shape) of M units. If someone gives you a value for M, determine the shortest possible length of a side.
this is a problem my teacher asigned us for the wekkend i have worked on it for two hourrs but nothing is making sense. im not sure what to do at all. PLEASE HELP ME
Found 2 solutions by checkley77, scott8148:
Answer by checkley77(12844)   (Show Source):
You can put this solution on YOUR website! YOU DIDN'T STATE THAT THIS TRIANGLE IS A RIGHT TRIANGLE OR NOT.
IT DOES MAKE A DIFFERENCE.
PERIMETER (M)=A+B+C FOR ALL TRIANGLES.
LET M=12
A RIGHT TRIANGLE COULD BE:
3+4+5=12
BECAUSE 3^3+4^2=5^2
9+16=25
25=25 THUS THE SMALLER SIDE WOULD BE 3.
HOWEVER A NON-RIGHT TRIANGLE WOULD HAVE A DIFFERENT SOLUTION GIVEN AN (M) VALUE.
TRY (M)=47
THE SHORTEST SIDE COULD BE 1.
A+B+C=47
1+B+C=47
1+23+23=47 (THIS TECHNIQUE WORKS AS LONG AS THE DIFFERENCE BETWEEN THE B & C SIDES IS < THE A SIDE.)
47=47
THIS WORKS FOR ANY PERIMETER = AN ODD NUMBER.
LETS TRY AN EVEN NUMBER:
(M)=74
A+B+C=74
2+B+C=74
2+36+34=74 (AGAIN THE DIFFERENCE BETWEEN B & C IS < A SIDE.)
74=74
Answer by scott8148(6628)  (Show Source):
You can put this solution on YOUR website! the sides are all whole numbers (so M must also be a whole number)
the smallest (non-zero) whole number is 1
in a triangle, the sum of any two sides must be greater than the third side; otherwise the triangle doesn't "close up"
so 3 is the smallest possible value for M ___ with a side of 1
M=4 doesn't close ___ 1+1 is NOT greater than 2
M=5 ; 2, 2, 1
M=6 ; 3, 2, 1 doesn't close ___ but 2, 2, 2 works
M=7 ; 3, 3, 1
M=8 ; 4, 3, 1 doesn't close ___ but 3, 3, 2 works
it looks like:
___ if M is odd, then the shortest side is 1
___ if M is even (except for 4), then the shortest side is 2
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