2015 AMC 12B Problem 10

Below is the professionally curated solution for Problem 10 of the 2015 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

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Concepts:triangle inequality systematic listing

Difficulty rating: 1520

10.

How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?

$3$

$4$

$5$

$6$

$7$

Solution:

Let the distinct sides be $a \lt b \lt c.$ Since $a + b \gt c,$ the perimeter exceeds $2c,$ so $2c \lt 15$ and $c \le 6.$

The scalene triples with perimeter less than $15$ are $(6,5,3),$ $(6,5,2),$ $(6,4,3),$ $(5,4,3),$ $(5,4,2),$ and $(4,3,2).$ Of these, only $(5,4,3)$ is a right triangle, leaving $5.$

Thus, the correct answer is C.

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