2015 AMC 12B Problem 10

How many noncongruent integer-sided triangles with positive area and perimeter less than

15

are neither equilateral, isosceles, nor right triangles?

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Solution

Let the distinct sides be

a \lt b \lt c.

Since

a + b \gt c,

the perimeter exceeds

2c,

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.