Difference between revisions of "2003 AMC 12A Problems/Problem 7"

Revision as of 10:13, 11 November 2006

How many non-congruent triangles with perimeter $7$ have integer side lengths?

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$

By the triangle inequality, no one side may have a length greater than half the perimeter which is $\frac{1}{2}\cdot7=3.5$

Since all sides must be integers, the largest possible length of a side is $3$

Therefore, all such triangles must have all sides of length $1$ , $2$ , or $3$ .

Since $2+2+2=6<7$ , atleast one side must have a length of $3$

Thus, the remaining two sides have a combined length of $7-3=4$ .

So, the remaining sides must be either $3$ and $1$ or $2$ and $2$ .

Therefore, the number of triangles is $2 \Rightarrow B$ .

Revision as of 22:43, 9 November 2006 (view source) Xantos C. Guin (talk \| contribs) (added problem and solution)	Revision as of 10:13, 11 November 2006 (view source) JBL (talk \| contribs) m (2003 AMC 12A/Problem 7 moved to 2003 AMC 12A Problems/Problem 7) Newer edit →
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